LC/6A,
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q /-3
AN EXPERIMENTAL AND THEORETICAL
STUDY OF DENSITY-WAVE OSCILLATIONS
IN TWO-PHASE FLOW
George Yadigaroglu
Arthur E. Bergles
Report No, .DSR74629-3
(HTL No. 74629-67)
Contract Nonr-3963(15)
ineering Projects Laboratory
Department of Mechanical Engineering
Mas sachus etts Institute of Te chnology
~imbridge, Massachusetts
02139
December 1969
ENGINEERING PROJECTS
,NGINEERING PROJECTS
GINEERING PROJECTS
7INEERING PROJECTS
~NEERING PROJECTS
-EERING PROJECTS
ERING PROJECTS
RING PROJECTS
ING PROJECTS
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PROJECT'
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LABORATORY
LABORATOR'
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11.
-11190
1111
AN EXPERIMENTAL AND THEORETICAL STUDY OF
DENSITY-WAVE OSCILLATIONS IN TWO-PHASE FLOW
George Yadigaroglu
Arthur E. Bergles
prepared for
Power Branch
Office of Naval Research
Department of the Navy
Washington, D.C.
under
Contract Nonr-3963 (15)
Engineering Projects Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Reproduction in whole or in part is permitted by the U.S. Government
Distribution of this Document is Unlimited
Report DSR 74629-3
ERRATA
p. 39
Title of Table 2.3: replace Section4
p. 62
16th line: replace %eqiAed by speciiled
p. 139
2nd line: replace boiing channet
Fig. 3.4 Ordinate should be
by Lengths
inge-phue tegion
by
6 instead of 66
Fig. 4.1 Solid lines refer to UNIFORM and dotted lines to COSINE
heat flux distribution
Fig. 5.2 Labeling of curves in lower figure: Replace
and
argH*
by 27r - argQ* and 2ir - argH*
Fig. 5.6 4th line of caption: delete
Fig. 5.12b
Fig. 5.14
Labeling of the curves:
label
0.75
label
0.125 the unlabeled curve
the curve labeled
Label horizontal axes
Delete
0
'
Replace 6h
Replace
h
Delete the
Z
6Z'
from
Delete the
0.1.25
by 6 hob
Replace 6h
Fig. 5.15
Eq. (5.38)
bb
from 6 h b
by 6 hfbb
by
0
from
Sh'b
6
ZOb
Fig. 7.7
4th line of caption: mode2 should be mode
Fig. 8.2
Label 6W0 the arrow entering block
2
5
70
P2
argQ*
,
respectively
IIIII
III
I
Abstract
A theoretical and experimental investigation of fundamental aspects
of density-wave oscillations was undertaken. The experimental Freon-113
loop was operated at atmospheric pressure level with constant pressure
drop across a single test channel. A wide range of inlet temperatures,
flows and heat inputs was covered. The regions of unstable operation of
the channel were mapped for both uniform and "cosine" heat flux distributions. The effect of the cosine distribution was stabilizing. The
period of the oscillations was approximately equal to twice the "transit
time", defined as the sum of one half the residence time of a fluid
particle in the single-phase region plus the vapor transit time in the
two-phase region.
At high subcoolings and low power levels unexpected "higher-order"
oscillations were detected. These were characterized by frequencies
that were multiples of the expected values. When the regions of unstable
operation were classified according to the "order" of the oscillation, a
coherent stability map emerged. The order of the oscillation is related
to the number of nodes in the standing enthalpy waves generated by oscillating flow in the single-phase region.
Analytically the stability of the channel is investigated by
oscillating the inlet flow. The linearized dynamics of the single-phase
region account for heat storage in the channel walls and for the effects
of the static and dynamic pressure variations on the movements of the
boiling boundary. The description of oscillatory two-phase flow is
Lagrangian and distributed in space. Considerable simplification in the
calculation of the pressure drop is achieved via subdivision of the channel
into intervals separated by points of constant enthalpy. The space variations of the fluid properties are taken rigorously into account while the
variations of the saturation enthalpy and the relative velocity between
the phases are treated approximately.
The theoretical procedures used for predicting the steady-state
condition in the boiling channel were tested by extensive pressure drop
measurements. The predictionsof the stability model are compared to the
experimental observations.
53 5220
Acknowledgements
This investigation was sponsored by the Power Branch
of the Office of Naval Research under Contract Nonr-3963 (15).
Thanks are due to Professors E.A. Mason, P. Griffith and
H. Fenech for discussion of various aspects of this work.
The Engineering Projects Laboratory staff gave technical
assistance.
The machine computations were performed at the
M.I.T. Information Processing Center and at the Mechanical
Engineering Department Computer Facility.
Acknowledgement is
also due to Miss Anne Hsu for her diligent typing of the
manuscript.
InNIM00"pp"
- ------""NOPWAN""M
I -
-...
Iwwlmw
IN1111
- 5 -
TABLE OF CONTENTS
Pag
List of Figures
10
List of Tables
13
Nomenclature
14
Chapter 1:
INTRODUCTION AND SUMMARY
1.1 Definition of the Problem
1.1.1 Density-Wave Oscillations - The Physical
Phenomenon
19
20
22
Density-Wave Oscillations in Boiling Water
Reactors
24
Previous Work at M.I.T.
25
1.4 Experimental Work
1.4.1 Higher-Mode Oscillations
26
27
1.5 Analytical Models
1.5.1 Single-Phase Region
1.5.2 Two-Phase Region
1.5.3 Stability Criterion
28
30
31
32
1.2
1.3
EXPERIMENTAL APPARATUS
34
2.1 Description of the Loop
2.1.1 Heated Test Section Assembly
2.1.2 Operating Range
35
38
40
2.2 Instrumentation
2.2.1 Pressure Measurements
2.2.2 Flow Measurements
2.2.3 Electric Power Measurements
2.2.4 Temperature Measurements
2.2.5 Other Measurements
41
41
42
44
44
45
Chapter 2:
Chapter 3:
3.1
PREDICTION OF THE STEADY-STATE OPERATING
CONDITIONS IN THE BOILING CHANNEL
General Calculation Procedure
3.2 Pressure Drop
3.2.1 Frictional Pressure Drop
3.2.1.1 Single-Phase Region
3.2.1.2 Two-Phase Region
46
47
48
49
49
50
-6Page
3.2.2
3.2.3
Gravitational Pressure Drop
Acceleration Pressure Drop
51
51
3.3
The Segment Containing the Bulk Boiling Boundary
52
3.4
Prediction of the Void Fraction
53
3.5
Void Fraction in the Subcooled Boiling Region
54
3.6
Temperature Profile
56
3.7
Heat Losses
56
3.8 Parametric Study of the Pressure Drop
3.8.1 Effects of the Inlet Temperature and the
Mass Flow Rate
3.8.2 Relative Importance of Friction, Gravity
and Acceleration Terms
Chapter 4:
4.1
MEASUREMENT OF STEADY-STATE PRESSURE AND TEMPERATURE
PROFILES AND COMPARISON WITH PREDICTIONS
57
57
58
59
60
Experimental Procedure
4.2 Measured Pressure and Temperature Profiles
4.2.1 Total Pressure Drop at the Threshold of Stability
61
62
4.3 Accuracy of the Predictions
4.3.1 Pressure Drop Predictions
4.3.2 Temperature Predictions
62
62
65
4.4
Effect of Subcooled Boiling
66
4.5
Choice of the Void Fraction Correlation
66
4.6
Observed Hysteresis in
4.7
Effect of the Heat Flux Distribution on the
Pressure Drop
4.8
The Pressure Drop in
Piping
4.9
Conclusions
Chapter 5:
5.1
the Pressure Drop Characteristic-
67
70
the Unheated Entrance
DYNAMICS OF THE SINGLE-PHASE REGION
Dynamics of the Heated Wall
5.2 The Flow-to-Local-Enthalpy Transfer Function
5.2.1 Solution for Arbitrary Heat Flux Distribution
5.2.1.1 Non-Dimensional Form'of the Transfer Functions
5.2.1.2 Characteristic Constants
5.2.1.3 High Frequency Approximation
70
71
73
75
80
80
84
85
86
IMMINIMN,
- 7 Page
5.2.1.4
5.2.1.5
5.2.2
Solutions for Given Heat Flux Distributions
5.2.2.1
5.2.2.2
5.2.2.3
5.2.3
91
91
Understanding the Physical Phenomena A Lagrangian, Non-Linear Solution
An Exact Non-Linear Lagrangian Solution
The Position of the Boiling Boundary
5.3.1
5.3.2
Effect of the Enthalpy Variations
Effect of the Pressure Variations
5.3.2.1
5.3.2.2
5.3.3
Static Pressure Effects - Variable Saturation
Enthalpy Along the Channel
Dynamic Pressure Effects
The Flow-to-Position-of-the-Boiling-Boundary
Transfer Function
5.3.3.1
5.4
Remarks
The Pressure Drop in the Single-Phase Region
5.4.1
The Equivalent Inertia Length
DYNAMICS OF THE TWO-PHASE REGION - THE LAGRANGIAN
ENTHALPY TRAJECTORY MODEL
Chapter 6:
Introduction
Derivation of the Basic Equations
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
6.2.6
89
89
90
Importance of the Wall Heat Storage
Importance of the Unheated Portions of
the Wall
Uniform Versus Cosine Power Distribution
Effects of Channel Segmentation
5.2.4.1
5.3
88
90
5.2.3.3
5.2.3.4
6.1
6.2
Uniform Heat Flux Distribution
Exact Chopped Cosine Heat Flux Distribution
Stepwise Varying Heat Flux Distribution
87
88
Heat Flux Distribution and Wall Heat Storage
Effects
5.2.3.1
5.2.3.2
5.2.4
Case of Negligible Heat Storage in the
Wall - Low Frequency Approximation
Case of Highly Conductive Wall
The Continuity Equation
The Energy Equation
Summary of the Solutions
First-Order Approximation
Zero-Order Approximation
A Remark on the Character of the Model
92
92
93
95
98
98
98
99
99
100
102
102
103
105
105
107
109
111
114
115
116
117
- 8 Page
6.3
118
Calculation of the Pressure Drop
119
119
121
6.4 Computation Procedure
6.4.1 Reference Steady State
6.4.2 Time-Dependent Hydrodynamic Calculations
6.5
Chapter 7:
124
Comments on the Method
PRESENTATION AND DISCUSSION OF THE STABILITY
EXPERIMENTS
128
129
130
7.1 Experimental Procedure
7.1.1 Blowdown Tests
7.1.2 Data Collection
7.2
7.3
7.4
126
Determination of the Threshold of Stability
Similarity Considerations and Non-Dimensional
Parameters
132
Data Reduction
136
134
7.5 Stability Maps
7.5.1 Higher-Mode Oscillations
7.5.2 Zero-Order Stability Boundaries - Uniform
Heat Flux Distribution
137
138
7.6 Period of the Oscillation
7.6.1 Time Delays in the Channel
7.6.2 Period of the Oscillation at the Threshold of
Stability
7.6.3 Period of the Oscillations in the Unstable
Region
143
143
7.7
Cosine Versus Uniform Heat Flux Distribution
146
7.8
Oscillation Amplitude in the Unstable Region
The Limit Cycle
7.9
STABILITY ANALYSIS
144
145
-
Experiments to Confirm the Mechanism of HigherOrder Oscillations
7.10 Summary and Conclusion
Chapter 8:
142
147
147
149
150
8.1 Stability Criterion
8.1.1 Range of Frequencies
151
152
8.2 Computer Program for Stability Analyses
8.2.1 Exit Pressure Perturbations
8.2.2 Phase of the Pressure Perturbations
8.2.3 Options and External Corrections
8.2.4 Performance of the Code
153
154
155
155
156
8.3 Prediction of the Threshold of Stability
8.3.1 Discussion of the Discrepancies
8.3.2 Higher-Mode Oscillations and Transitions
156
157
161
110
111iki
-
9 Page
Chapter 9: GENERAL CONCLUSIONS AND RECOMMENDATIONS
9.1
Steady-State Work
162
162
9.2 Stability Work
9.2.1 Experimental Results
9.2.2 Theoretical Results
163
163
164
9.3 Recommendations
9.3.1 Experimental Work
9.3.2 Analytical Work
166
166
166
REFERENCES
168
Appendix A: PRESDROP, a Computer Code to Predict the SteadyState Operating Conditions in the Boiling Channel
Al
Appendix B: The Physical Properties of Freon-113
Bl
Appendix C: Auxiliary Functions for Pressure Drop Calculations
* Cl
Appendix D: Steady-State Pressure Drop Data
Dl
Appendix E: Stability Data
El
Appendix F: Enthalpy Trajectory Model (Computer Program)
Fl
FIGURES
Distribution List
-
10
-
LIST OF FIGURES
Page
Fig.
2.1
Schematic of Apparatus with Overall Dimensions
2.2
Coding of Loop Equipment and Test Section Dimensions
2.3
Cross-Sectional Detail of Test Length and Coupling
2.4
Flow Measuring Instrumentation
3.1
Comparison of Void Fraction Correlations
3.2
Point of Net Vapor Generation as Predicted by Two Models
3.3
Effect of the Inlet Temperature on Various Parameters
3.4
Effect of the Mass Flow Rate on Various Parameters
3.5
Relative Importance of the Friction, Gravity and
Acceleration Terms. Contour Map at 792 Btu/hrft
4.1
Predicted and Measured Total Pressure Drop
at q" = 4700 Btu/hrft 2
4.2
Predicted and Measured Total Pressure Drop
at q" = 9400 Btu/hrft 2
4.3
Predicted and Measured Total Pressure Drop
at q" = 14100 Btu/hrft 2
4.4
Pressure Profiles at 4700 Btu/hrft 2 , Cosine Heat Flux
Distribution
4.5
Pressure Profiles at 9400 Btu/hrft 2 , Uniform Heat Flux
Distribution
4.6
Pressure Profiles at 14,100 Btu/hrft 2 , Uniform Heat Flux
Distribution
4.7
Accuracy of the Pressure Drop Predictions for All
Threshold and Transition Points
4.8
Pressure Drop Hysteresis
4.9
Temperature Profiles for Points A and B of Fig. 4.8
5.1
The Heated Wall
5.2
Flow-to-Local-Enthalpy and Flow-to-Heat-Flux Transfer
Functions at z = 2 ft
5.3
Definition of the Delays as Used in this Work
76
85
1=0110101INNOW11611
ill,114
1111AINHIMIM
N111.1161,
-
11
-
Page
Ig
5.4
Parametric Study of the Enthalpy Perturbation at the
Boiling Boundary, 6 hb
5.5
Chopped Cosine Heat Flux Distribution
5.6
Flow-to-Local-Enthalpy Transfer Function Along the
Channel. Zero-Order Oscillation
5.7
Flow-to-Local-Enthalpy Transfer Function Along the
Channel. First-Order Oscillation
5.8
Flow-to-Local-Enthalpy Transfer Function Along the
Channel. Fourth-Order Oscillation
5.9
Comparison of the Flow-to-Local-Enthalpy Transfer
Functions for Uniform and Cosine Heat Flux
Distributions. Zero-Order Oscillation
5.10
Comparison of the Flow-to-Local-Enthalpy Transfer
Functions for Uniform and Cosine Heat Flux
Distributions. Fourth-Order Oscillation
89
5.11 Effect of the Channel Segmentation on the Flow-to-LocalEnthalpy Transfer Function
5.12
Enthalpy Perturbations Along the Channel, 6V/Vo = 0.1
5.13
Enthalpy Perturbations Along the Channel, 6i//Vo = 1
5.14
Effect of the Pressure on the Movement of the
Boiling Boundary
5.15
Effect of the Dynamic Pressure Variations on the
Movement of the Boiling Boundary
6.1
Definition of Coordinates
108
6.2
Time-Independent Reference Conditions in the Boiling
Channel
119
6.3
Trajectories of the Center-of-Volume (and Constant-Enthalpy)
122
Planes
7.1
Stability Map at 100 W/TL, Uniform Heat Flux Distribution
7.2
Stability Map at 200 W/TL, Uniform Heat Flux Distribution
7.3
Stability Map at 300 W/TL, Uniform Heat Flux Distribution
7.4
Stability Map at 100 W/TL ave., Cosine Heat Flux Distribution
7.5
Stability Map at 200 W/TL ave., Cosine Heat Flux Distribution
7.6
Recordings of Oscillations - Transition Points
-
12
-
Page
Fig.
7.7
First Occurence of Instabilities and Fundamental Mode
Oscillation Boundaries
7.8
Stability Boundaries for Uniform Power Distribution,
Zero-Order Points Only
7.9
Stability Boundaries for Cosine Power Distribution Zero-Order Points Only
7.10
Stability Map in the Dimensionless Enthalpy Pressure Drop Plane - Zero-Order Points Only
7.11
Period of the Oscillations at the Threshold of
Stability - All Orders at 200 W/TL
7.12
Period of the Oscillation at the Threshold of Stability Zero-Order Points Only - Uniform Heat Flux Distribution
7.13
Period of the Oscillation at the Threshold of Stability Zero-Order Points Only - Cosine Heat Flux Distribution
7.14
Period of the Oscillation Within the Unstable Region
(Contour Map)
7.15
Enthalpy and Single-Phase Delays - Uniform Versus Cosine
Heat Flux Distribution
7.16
Typical Oscillation Recordings
7.17
Stability Map in the Enthalpy - Subcooling Plane E Series Experiments
7.18
Stability Map in the Enthalpy - Single-Phase-Length
Plane - Series E Experiments
8.1
Open Loop Block Diagram of the Boiling Channel
8.2
Closed Loop Block Diagram of the Boiling Channel
8.3
Correlation of the Period of the Oscillation
8.4
Open Loop Transfer Functions at Low Subcooling
8.5
Open Loop Transfer Functions at High Subcooling
8.6
Vector Diagram of Perturbations as Typically Predicted
by the Model
157
8.7
Corrections to Predicted Perturbations
158
WINW40
Pill"
-
13
-
LIST OF TABLES
Table
Page
2.1
Characteristics of the Experimental Loop
36
2.2
Selected Physical Properties of Freon-113
37
2.3
Physical Properties of the Test Sections
39
2.4
Operating Range of the Independent Variables
40
4.1
Definition of the Heat Flux Distributions Used Throughout
This Work
59
Accuracy of the Total Pressure Drop Predictions at the
Threshold of Stability and the Transition Points
63
Effect of the Relative Velocity Between the Phases (As
Predicted by the L-M Correlation) on the Gravitational
and Acceleration Pressure Drop
68
4.2
4.3
7.1
C.1
Range of the Independent Variables During the Stability
Experiments
Range and Accuracy of the Numerical Fits of the Auxiliary
Functions
128
187
-
14 -
NOMENCLATURE
Symbols which occur only infrequently and are explained in
immediate context have not been included in this list.
A coherent
system of units is assumed; the conversion factors that are necessary
when British units are used were not included in the formulas.
Latin Alphabet
A
flow area
A(z)
coefficient, Eq. (5.29)
a
coefficient defined by Eq. (5.9)
B
coefficient,
Eq.
(5.30)
C
coefficient,
Eq.
(5.13)
c
specific heat of the liquid
D
inside diameter of the channel
d
thickness of the channel wall
E(z)
coefficient, Eq. (5.14)
F(s)
complex expression, Eq. (5.20)
f
Fanning friction factor
G
mass flux
g
acceleration of gravity
H(z,s)
flow-to-local-enthalpy transfer function, Eq. (5.34)
h
enthalpy
h
forced convection heat transfer coefficient (Chapter 5)
d1l"
-
15
-
h
latent heat of vaporization
j
imaginary unit
j
volumetric flux density (Chapter 6)
j
volumetric flux density in the moving frame of
reference (Chapter 6)
K(s)
complex expression, Eq. (5.32)
k
thermal conductivity of the wall
Lh
length of the channel from Station 1 to exit, Fig. 2.2
L(s)
complex expression, Eq. (5.33)
It
length
P
heated perimeter of the channel
Pr
Prandtl number
Plw(s)
flow-to-pressure-drop-in-the-single-phase-region
function, Eq. (5.51)
Plz
position-of-the-boiling-boundary-to-pressure-drop-in-thesingle-phase-region transfer function, Eq. (5.50)
p
pressure
Q(z,s)
flow-to-heat-flux transfer function, Eq.
q
total heat input rate
q.
heat input rate per test length (i = 1,7)
q'
linear heat input rate
q"Iheat
(5.36)
flux
q'''
volumetric power density
Re
Reynolds number
s
complex frequency, Laplace variable (Chapter 5)
s
slope, Section 6.2.4
transfer
-
16 -
T
temperature
t
time
V
velocity
v
specific volume
vfg
v
w
mass flow rate
Z(s)
flow-to-position of-the-boiling boundary transfer
function, Eq. (5.48)
z
axial coordinate, measured from Station 1, Fig. 2.2
z'
axial coordinate, measured from exit, z' = Lh - z
zbb
coordinate of the boiling boundary, measured from
fg
W(s)
- vf
gf
open loop transfer function, Chapter 8
Station 1
z in
equivalent inertia length, Section 5.4.1
Greek Alphabet
a
void fraction
thermal diffusivity (Chapter 5) of the wall
coordinate in the moving frame of reference, Section 6.2
A
wavelength, Section 5.2.1.2
viscosity
p
average density of the liquid-gas mixture, p a + p(-
a
surface tension
T
period of the oscillation
Tc
time constant, Eq. (5.39)
-
17
-
Th
convective time constant, Eq. (5.38)
Tk
2
wall conduction time constant, Eq. (5.37)
#t
Lockhart-Martinelli, two-phase friction multiplier
(relative to liquid phase)
2
#
gtt
Lockhart-Martinelli two-phase friction multiplier
(relative to gas phase)
SI
coefficient, Eq. (6.3)
W
angular frequency,
2w/T
Operators
d
differential operator
a
partial derivative
A
difference
6
small perturbation
sum
average
Subscripts
ac
acceleration
b
bulk (mixed-mean)
bb
boiling boundary
d
point of net vapor generation, Section 3.5
e
segment exit
ex
channel exit
f
liquid phase
f
feedback (Chapter 8)
-
18
-
fg
transition from liquid to vapor
fr
friction
g
gas phase
gr
gravity
in
inlet of segment
i
enthalpy mesh point (Chapter 6)
in
inlet of heated section
in
inertia (Chapters
5 and 8)
liquid phase
sat
saturation
subb
subcooling with respect to the boiling boundary,
Section 7.4
suin
subcooling with respect to the heated channel inlet,
Section 7.4
0
average value at inlet (Station 0 or 1)
1
single-phase region
2
two-phase region
00, 0, 1,
.,
8,
ex
Stations along the channel,
Supercripts
center-of-volume-plane index, Section 6.4.2
reference steady state
dimensionless quantity
peak amplitude
-~-~
Fig. 2.2
I
lilllMillnbl
ildllel
~~ ~ ~
g ,Il||11,1
unlII,
-
19
-
Chapter 1
INTRODUCTION AND SUMMARY
The study of instabilities occurring in two-phase flow systems was
initiated approximately thirty years ago.
Greatly increased attention
was given to these problems in the last fifteen to twenty years with the
advent of high power density boilers and boiling water reactors
(BWR).
Review articles that appeared in this period follow the historical
developments [1-6]*.
The particular subject of this work, the so-called "density-wave
oscillations," will be outlined in this introductory chapter within the
larger framework of two-phase stability phenomena.
A brief preview of
the subsequent chapters is given in Sections 1.4 and 1.5.
Chapter 2 describes the experimental apparatus.
Chapters3 and 4
deal with the prediction and the measurement of the steady-state
conditions in the channel.
They need little introduction and can be
left aside without loss of continuity by the reader more interested in
the stability problems.
phase region.
Chapter 5 treats the dynamics of the single-
A model for the dynamics of the two-phase region is devel-
oped in Chapter 6. The stability experiments are described in Chapter 7.
The dynamic models developed in Chapters 5 and 6 are then assembled in
Chapter 8 to predict the threshold of stability.
The work is concluded
in Chapter 9. Each chapter can be read independently as cross references
*
Numbers in square brackets designate references listed in pages 168-174
-
20
-
provide the necessary links to other sections.
1.1
Definition of the Problem
It is first necessary to make the distinction between microscopic
instabilities which occur locally at the liquid-gas interface, for
example
etc.,
the well known Helmholtz and Taylor instabilities, bubble collapse,
and macroscopic instabilities which involve the entire two-phase
flow channel.
Acoustic waves, shock waves, and critical flow phenomena
constitute another important family of two-phase dynamics problems characterized by relatively high frequencies.
Only macroscopic channel
instabilities, involving relatively slow transients (time constants of the
order of a few seconds) will be considered here.
These can be essentially
categorized as instabilities of flow distribution among several parallel
channels and flow instabilities in a single channel.
The first, is beyond
the scope of this work which deals only with one important class of singlechannel flow instabilities.
The major distinction between single and
parallel channels stems from the formulation of the boundary conditions.
For example, consider a BWR core.
To study the redistribution of flow
among the subassemblies in the case of some large external disturbance, it
is necessary to examine the entire primary loop.
On the contrary, when
the stability of the hot channel alone is investigated, it is sufficient
to specify the pressure at the inlet and exit plenums, as flow changes
in a single channel will not affect significantly the total flow, and
consequently the pressure drop, across the core.
Therefore, whenever
the conditions at the channel limits are sufficiently specified at all
-
21
-
times, the problem reduces to a single-channel instability.
There has been considerable confusion in the past as many experiments
were designed without making an effort to clearly define the boundaries
of the unstable part of the system. In particular, system-induced instabilities have often interfered
with more fundamental oscillation modes.
Furthermore, the term "parallel channels" has been used frequently to
recall a constant-pressure-drop boundary condition, while it might have
been used more properly to denote flow distribution phenomena.
It is clear that it is necessary to solve first the stability
problems in a single channel before attempting to synthesize the parallelchannel solutions.
The discussion that follows is limited to single
channels.
The thermo-hydrodynamic instabilities occurring in a boiling channel
can be classified in several ways.
According to the circulation mode, it
is possible to make the distinction between instabilities occurring with
natural and forced convection.
Considering, however, the similarity of
the appropriate boundary conditions this distinction is removed.
Only
the range of certain parameters will differ in the two cases.
A more fundamental approach would be to characterize the instabilities
by the nature of the phenomena taking place.
explicitly done so in
the past.
Few investigators have
The general term of "two-phase flow
instabilities" was used for dissimilar phenomena.
Recent publications
[5,7] have finally brought some order and clarity, and, most importantly,
categorized the various kinds of possible instabilities.
The distinction will be made first between static or excursive and
-
dynamic or oscillatory instabilities.
classical example of the first type.
22
-
The Ledinegg instability is the
Stenning and Veziroglu [7],
in their
experiments with Freon-li, have encountered two major modes of dynamic
instabilities, which they have termed "density-wave oscillations" and
"pressure-drop oscillations."
Their findings are believed to be
representative of the general situation.
They have also reported the
occurrence of "thermal oscillations" in their experimental apparatus,
and postulated a mechanism for these based on a relatively rare combination of flow and heat transfer characteristics.
Instabilities generated
by particular phenomena (flow regime transitions, nucleation, etc.) that
for some reason acquire a major importance under unusual conditions, for
example
these thermal oscillations, seem to defy a priori classification.
Therefore, the list of instability mechanisms should be left open to
include these minor types.
Pressure-drop oscillations occur (like the Ledinegg instability)
when the pressure drop across the channel decreases with increasing flow.
They contrast, however, the Ledinegg instability by their oscillatory
nature.
Their frequency is
of the system,
governed by the volume and compressibility
and they occur in
the low exit quality region only, where
the pressure drop - flow rate characteristic exhibits a negative slope.
1.1.1
Density Wave Oscillations - The Physical Phenomenon
The density-wave oscillations that constitute the subject of this
work are due to the multiple regenerative feedbacks between the flow rate,
the vapor generation rate, and the pressure drop.
Inlet flow fluctuations
11lllimbI
IMMURE
IN
MIN
1111161
-
create enthalpy perturbations in
23
-
the single-phase region.
When these
reach the boiling boundary they are transformed into void fraction
perturbations that travel with the flow along the channel, creating a
dynamic pressure drop oscillation in the two-phase region.
Since the
total pressure drop is imposed upon the channel externally, this twophase pressure perturbation produces a perturbation of the opposite sign
in the single-phase region, which in turn creates further inlet flow
variations.
It is
evident that with correct timing,
the perturbations
can acquire appropriate phases and become self sustained.
Therefore,
transportation delays in the channel are of paramount importance for the
stability of the system, although inertia effects are also responsible
for the generation of phase shifts.
This led to the suggestion that
these oscillations be called "time-delay" oscillations.
Emphasizing the
feedback mechanism, Neal and Zivi [5] named them "flow-void feedback
instabilities".
Oscillation of any physical system requires a mechanism for intermittent storage and release of some particular form of energy.
In the
present case, for constant heat input, the total energy contained within
the channel varies according to the mass flow rate.
At high flow rates
little energy is stored; when the flow diminishes the amount of energy
contained in the channel increases.
Transient heat storage in the channel
walls, variation of the saturation temperature with pressure, compressibility effects, and thermal non-equilibrium constitute auxiliary energy
storage and release mechanisms, that can create additional phase shifts.
Many other secondary phenomena also contribute to the fundamental
feedback mechanism described above, e.g.: variable heat transfer, flow
-
24
-
regime changes, the relative velocity between the phases, etc.
Although
none of these is the cause of density-wave oscillations, they might acquire
under certain conditions a controlling role.
The class of instabilities considered here will be further specialized
by the boundary condition that the pressure drop across the boiling
channel is maintained constant.
This is approximately the case when the
pressure drop-flow rate characteristic of the system is sufficiently
"stiff."
The boundary condition is even closer approximated in the most
unstable channel (probably the hot channel) of a large array of parallel
channels.
Furthermore a natural circulation loop can be described in this
fashion if the cold leg is included as part of the channel [5].
1.2
Density-Wave Oscillations in Boiling Water Reactors
The safety and operational evaluation of the
BWR's
provided the
main incentive for the numerous two-phase flow stability studies.
Although
the first investigators attempted to explain the flow instabilities in
BWR's
in terms of the rather unique reactivity-void coupling, e.g. [8],
it was soon realized that these phenomena could usually be accounted for
by purely hydrodynamic considerations.
The void-reactivity feedback
plays an incidental, but contributing, role in the overall reactor
stability.
It is now generally believed that the thermo-hydrodynamics can
be profitably studied independently of the dynamics of the reactor.
this premise several theoretical investigations [9-18] and test loop
stability experiments [19-24] followed.
On
-
25
-
BWR's operated at relatively low pressure levels, and
The early
consequently their power density was limited by flow stability considerations.
The large commercial
BWR's
built or planned in the United
States in the late sixties, operate at higher pressures and have overcome
Their thermal output is limited by the critical
the stability problems.
heat flux ratio.
There is, however, no assurance that new, economically
improved designs will not bring back the concern about the thermo-hydrodynamic stability.
Futhermore, flow stability is an important consideration
in the design of pressure-tube reactors and will probably be the limiting
factor in the design of large scale low-pressure desalination plants.
The typical
BWR
has a closed box subassembly.
It stands therefore
between the open lattice core and the single tube channel.
It is suspected
that two-dimensional effects in the rod bundle do somehow alleviate the
stability problems.
No theoretical work has been reported on the
stability of rod bundles.
This is not surprising as the difficulties of
predicting even the steady-state operation are considerable.
The reactor
designers have to rely on full-scale tests of pressure tubes and rod
bundles to verify their stability, e.g. [25,26].
1.3
Previous Work at M.I.T.
A number of previous investigations prepared the ground for the
present work.
Steady-state experiments with fluorocarbons in heated
glass tubes provided the necessary experience with such systems [27,28].
Early stability experiments [29] gave very valuable indications and guided
the conception of the present experiment.
Extensive surveys of the two-
phase flow literature [30] and a detailed survey of the stability
-
problems
26
-
[31] provided for the collection and classification of the
background information.
Crowley, Deane and Gouse [32] reported the first experimental results
obtained with the present equipment.
The experimental part of this
investigation is essentially an expansion and completion of their work.
1.4
Experimental Work
The primary goal of the present experimental work was to produce a
complete stability map that could be used to test various analytical
models.
The experimental facility, described in Chapter 2, was designed
[33] to produce density wave oscillations under well defined conditions,
without interference from other kinds of instabilities.
The constant-
pressure-drop boundary condition was assured by a large bypass in
parallel with the test channels and a sufficient condenser volume.
Although the facility had three vertical, identical parallel test channels,
all the experiments were run, as described in Chapter 7, with a single
channel in order to avoid any possible coupling between channels.
In practice, the independent variables were easily controllable;
thus the system could be operated at any point within its range.
This
contrasts the experiments conducted with natural circulation, e.g.
[19,20,21,23], where only certain curves could be followed in the operational space.
With Freon-113 as the test fluid, the experiments could be pursued
safely inside the unstable region without encountering excessive heated
wall temperatures.
The instabilities where thus dissociated from the
-
27 -
burnout phenomena, and valuable information on the limit cycle could
be gathered.
In many similar experiments a primary fluid loop was used to
transfer heat to the test channel.
The use of easily controllable elec-
tric heating in the present experiment eliminated potential coupling of
the test channel dynamics with the primary heat source dynamics.
Glass was used extensively to permit visualization of the flow.
Visual observations of the oscillating test channels provided valuable
insight into the problem.
Operation at atmospheric pressure levels
emphasized phenomena that would have been unimportant at higher pressures.
The effect of the axial heat flux distribution has been recognized
earlier [9,20] but very little experimental data areavailable.
Segment-
ation of the heated test sections permitted simulation of non-uniform
heat flux distributions and evaluation of their effect on the stability
of the channel.
1.4.1
Higher-Mode Oscillations
The occurrence of "higher-mode" oscillations and transitions from
mode to mode inside the unstable region was observed during the experiments, and is described in Section 7.5.1.
As no published reports on
such behavior could be found in the literature, part of the experimental
program was reoriented to examine closely these phenomena.
The higher-
mode oscillations were characterized by periods that were equal to a
fraction only of the expected period.
In fact, for density-wave oscil-
lations, it has been well established that the period is approximately
-
equal to twice the "transit time",
28
-
i.e. the time required for a fluid
particle to travel through the channel.
This can be intuitively under-
stood by considering the fact that the perturbations travel with the
flow and that the inlet flow perturbation is approximately
of phase with the pressure perturbations.
180*
out
Examination of the single-
phase dynamics (Chapter 5) and a series of ad hoc experiments (Section 7.9)
established that these higher modes were associated with the presence of
"standing enthalpy waves" in the single-phase region.
1.5
Analytical Models
Analytical investigations of stability in two-phase flow start with
the fundamental mass, momentum, and energy conservation equations [6,34],
augmented by semi-empirical relations such as heat transfer rate equations,
pressure drop correlations, etc. Obviously if enough detail is built into
a model, it should be possible to predict all excursions and oscillations.
This is, however, impractical in general, and the models must be specialized to suitably treat certain classes of problems.
Care must be exercised
then to include all features essential to the phenomena being considered.
The equations of the present problem are space and time dependent.
Direct integration of these is impossible unless drastic simplifications
are made.
The possible solutions can be classified according to the
methods used to eliminate the space and time dependences and arrive to a
stability criterion [37].
The models are termed lumped if the space
variable is eliminated by integration along the channel, assuming a priori
some approximate space dependence.
They are called distributed when the
01111,
29 -
-
In the time
space dependence is derived rigoiously from the equations.
domain, the equations are either linearized, to permit application of
the well-developed control theory, or treated numerically in a nonlinear fashion.
Although the notion of the boiling boundary, i.e. the point of the
channel where the mixed-mean enthalpy reaches saturation,
artificial,
it
is
it
rather
is
will be used extensively throughout this work.
In fact,
necessary to treat separately the single-phase and the two-phase
regions in
order to incorporate in
each formulation all the essential
features; the boiling boundary provides a convenient separation.
Moreover, a number of experimental observations can be explained by
considering the dynamics of the boiling boundary alone.
The existing analytical stability models fall into two categories.
In the first, the models, in taking into account all possible effects
evolve into complex computer codes, e.g. [12] or inextricable algebra,
from which little physical understanding can be extracted.
There are
also relatively cumbersome or expensive to use for preliminary survey
calculations.
In the second category, the models are reduced to a simple form,
which either limits their applicability or compromises seriously their
accuracy.
Credit must be given, however, to some models in this cate-
gory for contributing to the physical understanding of the stability
phenomena.
A middle course was taken here.
Fragmentation of the model into
smaller, independent blocks permitted retention of enough detail without
-
30
-
loosing touch with the physics of the problem.
The stability model
presented can be characterized as hybrid, in the sense that it makes use
of both linear, frequency-domain and non-linear, time-domain solutions.
As described in Chapter 8, given the steady-state conditions at some
point, the approximate frequency of a potential oscillation can be
obtained.
Indeed it is shown in Chapter 7 that the frequency of the
oscillation is uniquely correlated to channel variables.
of the channel is
frequency.
The stability
then tested by oscillating the inlet flow around this
For these calculations,
the single-phase region and the two-
phase region are dealt with separately.
Single-Phase Region
1.5.1
The single-phase region is treated in Chapter 5 in a rather conventional fashion; the equations are linearized and solved in
domain.
It
is
indeed generally agreed that linearized models predict
adequately the threshold of stability.
enthalpy with pressure,
The variations of the saturation
at the boiling boundary are taken into account,
while they were neglected in previous formulations.
however,
is
the frequency
Subcooled boiling,
ignored for lack of an adequate dynamic description.
The importance of heat storage in
the channel wall and variable heat
transfer have been recognized and the dynamics of the heated wall were
incorporated in
the formulations of the single-phase region.
As shown in
Section 5.2.3, the often neglected wall dynamics might play under certain
conditions an important role.
EEH
Him IiiII,
-
31
-
Solution of the equations of Chapter 5 yields the oscillations of the
boiling boundary and the variations of the single-phase pressure drop when
the inlet flow is
oscillated.
These two are then used as inputs to the
two-phase dynamics model.
Two-Phase Region
1.5.2
The treatment of the two-phase region is
novel.
A Lagrangian
description was used and exact distributed solutionswere obtained with
a minimum of simplifying assumptions.
Model"
is
The resulting "Enthalpy Trajectory
described in detail in Chapter 6.
Use of a Lagrangian approach
limits, however, the applicability of the model to channels with axially
uniform cross-section and heat flux.Variations of the heat transfer from
the wall are less important in the boiling region where the heat transfer
coefficient is weakly dependent on velocity and quality.
the model does not permit an accounting of these effects.
The structure of
Thermodynamic
equilibrium between the phases was assumed and, although the treatment of
the two-phase region is non-linear, the application of the equations is
confined to small and medium flow perturbations.
Flow reversals, for
example, cannot be dealt with.
The various existing models that deal with the dynamics of two-phase
flow are oriented toward high pressure systems as they do not take into
account the variations of the saturation temperature and other properties
with pressure in space.
Moreover, incorporation of time dependent
property variations increases the difficulty of obtaining numerical
solutions by an order of magnitude [35].
A compromise was reached here
by using a reference pressure profile which takes into account the space
variations of the properties.
This reference profile is provided exter-
-
32
-
nally with any available degree of accuracy.
The time dependence of
the saturation enthalpy is taken approximately only into account
(Chapter 8).
The stability model does not make use of the various semi-empirical
correlations required for the steady-state calculations but rather uses
a fundamental technique to determine the deviations from the reference
state.
The enthalpy trajectory model is used in Chapter 8 to predict
the pressure drop in the two-phase region with oscillating flow.
1.5.3
Stability Criterion
Once the pressure oscillations in
channel are
obtained,
the two distinct regions of the
use of the constant-pressure-drop boundary
condition allows the stability criterion to be written in
its simplest
form
6Ap 1 + 6Ap
2
+
0
the single-phase
where the two terms are the pressure perturbations in
and the two-phase regions, respectively.
In Chapter 8 this criterion is
put into a more convenient form for use in the stability analysis.
The procedure of predicting the threshold of stability outlinedabove
allows one to follow the calculations step by step.
Physical understand-
ing of the phenomena and the effect of parametric variations can be
easily extracted from the model.
As a test of the theory, a computer program was written to investigate the stability of the present experimental system.
In Chapter 8, the
theoretical predictions of this code are compared to the experimental data.
-
33
-
Although the stability model exhibited a qualitatively correct behavior,
the predicted threshold of stability was not in agreement with the
experimental observations.
As discussed in detail in Chapters 8 and 9,
effects that could not be included in the theory, for example lack of
thermodynamic equilibrium and radial heat transport effects in the
single-phase region, seemed to be mainly responsible for the discrepancy.
It is recommended that the model is further tested under different
experimental conditions, including high pressure water systems.
-
34
-
Chapter 2
EXPERIMENTAL APPARATUS
The basic experimental apparatus used in this work was described in
detail in previous publications that originated from the project
[32
- 33].
A brief description will be repeated here for completeness,
and numerous innovations that were made in the instrumentation will be
discussed.
The equipment evolved from previous designs [29], and was designed
with flexibility and simplicity in mind.
The vital part of the loop, the
test-section assembly, consists of three parallel heated glass channels
in parallel with a large bypass whose function is to maintain a constant
The test fluid enters the
pressure drop across the heated channels.
three channels at the bottom and flows, while evaporating, vertically
upward.
Use of glass in the loop permits visualization of the flow, but
limits the attainable pressure level.
Subdivision of each channel into seven individually powered test
lengths permitted simulation of non-uniform heat flux distributions,
similar to those encountered in boiling water reactors.
The rig was
designed for Freon-113, although other test fluids can be used.
Freon-113
was well suited for the test program since it has a boiling point of 118*F
at atmospheric pressure, a low vapor pressure, and a latent heat of
vaporization about one tenth that of water.
It is thus possible to
produce any exit quality and even superheated vapor with modest power, at
low temperature and without any fear of burnout.
Moreover, Freon-113 is
U111
-
35
-
relatively inexpensive, and exhibits the desirable properties of
fluorocarbons, i.e. it is non-corrosive, non-toxic, electrically nonIt has, however,
conductive, non-flammable, and chemically stable.
several potentially undesirable properties: a very low thermal conductivity, high air solubility, and high degree of wettability.
These
properties may significantly influence the heat transfer behavior
relative to that obtained, for example, with water.
2.1
Description of the Loop
Figures 2.1 and 2.2 are schematic diagrams of the loop.
Figure 2.1
describes in detail the hardware and general instrumentation, while
Fig. 2.2 gives exact test section dimensions and codes important
components.
The basic flow system consists of:
the pump, located well
below the entrance of the test sections to avoid cavitation; the main flow
lines; the pump bypass that permits isolation of the test sections; the
total flow metering lines; the main (HXl) and test section (HX2) preheaters; the test-section assembly in parallel with the large bypass;
the condenser and after-condenser; the downcomer; and the precoolers used
to lower the fluid temperature before reaching the pump.
Glass was
extensively used to permit visual observations of the boiling phenomena
and to monitor system performance.
Two preheaters were necessary in
order to be able to keep the temperature in the bypass below saturation,
while still independently varying the test-section inlet temperature.
The flow is divided between the pump bypass and the test sections
by adjusting valves Vl, V6 and V3 or V2.
The test-section bypass valve,
V9, controls the pressure drop across the test sections.
-
36
-
Two condensers were thought to be necessary to eliminate noncondensible gases from the main condenser; the idea was to maintain the
condenser pressure slightly above atmospheric and allow a slow leakage
of vapor to the after-condenser.
This, however, did not provide a
sufficiently constant exit pressure when the vapor generation rate was
varied during the experiments.
For this reason, after an initial degas-
sing, sufficient cooling was provided to the main condenser to keep the
pressure at atmospheric level (see also 4.1).
The condenser volume and
cooling were sufficient to eliminate any dynamic pressure variations.
The flow area in the test section preheater HX2 was large enough
so that there was essentially no frictional pressure drop across it for
the range of flows passed through the heated sections.
There was,
however, an inertial pressure drop in this leg (see 5.4.1).
The constant-
pressure-drop region therefore extended from point 00 (Fig. 2.2) or to a
lesser approximation point 0, to the channel exit (ex).
The heated
region between points 1 and 8, is described in more detail below.
Table 2.1 gives more general information about the loop.
Table 2.2
summarizes a few important properties of the test fluid for convenient
reference.
Table 2.1
Characteristics of the Experimental Loop
Construction Materials
Brass, copper and glass.
Pump
40-hp centrifugal pump, manufactured by Buffalo Forge.
"Chem-rated" AVS size 50.
Double mechanical carbon seals.
150 gal/min at 235 ft of water
head.
-011111
-
37
,
-
Table 2.1 Concluded
Heat Exchangers and Condensers
Tube and shell type, manufactured by American-Standard.
Cooling and heating by mixture
of steam and water.
Charge of Freon-113
Approximately 700 lbm.
Electric power to the test sections
Seven* General Radio Variac
W20G2 autotransformers,
220 V primary/0 to 330 V
secondary.
Total power available
Approximately 12 kW maximum
(limited by autotransformer
load).
* An eighth autotransformer could be used to power all seven Variacs at
reduced load in order to preserve the power distribution while changing
the total input.
Table 2.2
Selected Physical Properties of Freon-113 [41,42,43]
Chemical name and formula
trichlorotrifluoroethane, CCl 2F-CClF2
Boiling point
117.6*F at 14.7 psia
Liquid density
97.7 lbm/ft3
Critical Point
417.4 *F; 495.0 psia
Latent heat of vaporization
63.1 Btu/lbm at 14.7 psia
Specific heat of the liquid
0.213 Btu/lbm at 70 *F
Prandtl number of the liquid
7.3 at 100*F
Liquid-to-vapor density ratio
195 at 15.4 psia, 137 at 20.5 psia
See also Appendix
B
(1.565 g/cm ) at 77 *F
-
38
-
Heated Test Section Assembly
2.1.1
The test channels were made of Pyrex EC electrically conducting
glass tubing.
Seven test lengths, each 15-in. long, stacked and coupled
with 0.5-in. thick coupling plates, comprised the heated portion of a test
section.
Figure 2.2 gives the important dimensions and Fig. 2.3 shows
the details of the coupling.
The three test channels were connected to
a common header at the inlet, to a large transparent riser at the exit.
A short unheated length of copper tubing was provided at the end of
each test section to bring the channel exits above the liquid level in
the riser.
The liquid from the riser was directly drained to the down-
comer (Fig.
2.1),
while the vapor was passed to the condenser.
Struc-
tural rigidity and alignment were provided by three metal rods running
along the test sections, through the plastic couplings.
Two instrument taps were provided in each coupling for each of the
channels:
a 0.042-in. dia. pressure tap and a 0.062-in. dia. thermocouple
well.
The glass tubes were coated with a thin, optically transparent,
electrically
conductive film.
Heat was supplied to the tubes by passing
an electric current through this resistive coating.
The temperature
coefficient of resistivity of the coating was estimated to be of the
order of 10~4 /C.
Thus, in contrast with other investigations where
a second fluid was used to supply heat, the heat supply was essentially
constant in time (except for the negligible effect of time-varying heat
losses) and easily controllable.
Each Variac was powering all three
test lengths at one level; the power was equally distributed by means of
adjustable trimmer resistors.
U11,
-
39
-
The factory tin oxide coating was somewhat uneven resulting in a
rather consistent, slightly non-uniform axial resistance distribution.
Typically, the linear resistivity curve presented two humps situated at
1/3 and 2/3 of the total length.
The peak to average ratio was 1.15 for
the few tubes measured (1.33 if the unheated length is taken into
account).
The actual heat flux distribution would be slightly flattened
by axial heat conduction.
Electric power was supplied to the tubes by
wire braid power leads coiled around a heavier coating at each tube end.
the coupling plates and the power leads
The presence of the couplings,
leaves an unheated length of 2 to 2.5 in.
at each experimental station,
referred to in later chapters as cold spot.
The physical properties of the glass test sections are summarized
in Table 2.3.
Table 2.3
Physical Characteristics of the Test Sections
Geometry:
Overall length
Heated length
Diameters
Flow area
15 in.
13 in.
0.510 in. OD
0.0010085 ft
0.430 in. ID.
7740 Pyrex Glass (Corning
Glass Works)
Material
Physical Properties of the glass [44,45]:
3
density
162 lbm/ft
heat capacity
0.21 Btu/lbm*F at 250*F
thermal conductivity
0.75 Btu/hrft*F at 250*F
thermal diffusivity
0.022 ft 2/hr at 250*F
safe operating temperature
480*F
Conductive coating:
thickness.
approximately 16 x 10-6 in.
optical transparency
70 percent
total resistance
100 to 200 Q
-
2.1.2
40
-
Operating Range
As noted earlier, the pressure level is limited by the glass
portions of the loop.
The maximum heat input to the test sections is
given by the acceptable autotransformer load or the permissible safe
heat flux.
The mass flow rate was limited by the liquid evacuation
capability of the riser, which had an inadequate liquid drain; at exit
liquid flow rates in excess of 4000 lbm/hr flooding of the riser
occurred.
The temperature of the city water determined the lowest
inlet temperature obtainable (380 F during the coldest winter days).
The range of variables, as summarized in Table 2.4, was generally
sufficient for the present experiment.
Table 2.4
Operating Range of the Independent Variables
Pressure level (condenser)
0 to 20 psig
Maximum average* "safe" heat flux
(inside diameter)
16000 Btu/hrft #
corresponding to 680 W/test length
or to a linear heat rate of
1800 Btu/hrft
Flow rate (per tube)
up to 4000 lbm/hr
Inlet velocity
up to 12ft/s
Exit quality
subcooled liquid to superheated
vapor
Inlet subcooling
0 to 100 OF
2
* averaged including the unheated lengths at the experimental stations
successful operation at 50,000 Btu/hrft2 has been reported [27]
N
-
Ift
41 -
2.2 Instrumentation
The loop was equipped with a large number of instruments as shown
in Fig. 2.2.
Channel C, which was generally the only channel used, was
equipped with thermocouples and pressure transducers at all couplings.
Although the primary measuring elements (venturis, pressure transducers, and thermocouples) were the same for both the steady-state and
the stability runs, different recording instrumentation
was used for
each type of experiment. The millivolt transducer signals were measured
with a precision d.c. vacuum tube volt meter (Hewlett-Packard, Model
412A) during the steady-state runs.
A direct-writing oscillograph
(Visicorder, Model 906B, manufactured by Honeywell; 12 recording
channels) was used during the stability runs to record all the rapid
transients.
2.2.1
Pressure Measurements
Bourdon type precision pressure gauges were used to monitor the
condenser pressure (P95) and the pressure at the bottom of the test
sections (P93).
An open mercury manometer was connected to the inlet
header (Station 0). During the blowdown experiments (Section 7.1.1)
the level of the liquid in the graduated bypass was also used to determine
the inlet pressure.
Transducers were connected to the pressure taps at stations 0 to
8. A variety of "unbonded strain gauge" differential and gauge pressure
transducers were used. Their usable ranges extended from 1 psi to 30 psi.
They had been calibrated individually and their calibrations were often
checked using the liquid level in the test sections as a reference.
-
42
-
The reference side of the differential pressure transducers was
either connected to the condenser or left open to the atmosphere.
Occasionally, to extend the range of the differential transducers, a
liquid Freon head was applied as reference.
Regulated d.c. voltage excitation to the transducers and passive
signal adjustments were provided by a set of eight strain gauge bridge
adapters supplied by the Ramapo Instrument Co. (Model SGA-100B).
Whenever necessary, simple RC filters were installed between the bridges
and the recording instruments to eliminate the 60 Hz pickup.
During a single run, the total observed transducer drifts were
generally below 0.1 psi, typically between 0.03 and 0.1 psi, depending
on the range of the individual transducer.
This drift, caused by
thermals and introduction of vapor into sensing lines, was considered
in the data reduction.
Flow Measurements
2.2.2
Three venturis in the unheated inlet section of the channels
provided both the static and the dynamic flow measurements.
The calibra-
tion and characteristics of these units, designed to produce a minimal
pressure drop, are given in Ref. [32].
Pressure lines, approximately
one foot long, were used to feed the pressure signals to three
differential pressure transducers (± 1 psid),
in
parallel with three
inverted U-tube Freon manometers used for static measurements only.
Great care was exercised in eliminating trapped gas from the pressure
lines.
The transducer signal was read with the d.c. VTVM during the steadystate runs.
As this signal is approximately proportional to the square
- -16
-
43
&WIN11
I
-
of the flow, it was necessary to further process it in order to readily
interpret transient flow recordings.
An amplifier having a square-root
response was built to perform this function and is
Fig.
2.4.
shown schematically in
This unit had provisions for variable gain and convenient
calibration.
Unfortunately the pressure transducers available had e resonant
frequency in the vicinity of 10 Hz.
It was therefore necessary to follow
After a few trials,
the square-root amplifier by a RLC low-pass filter.
a satisfactory filter
having a 60 db/decade attenuation after a cutoff
frequency of 6 Hz (10 percent drop at 1.2 Hz,
60 percent at 5 Hz)
was
obtained.
Finally,
an integrating circuit was built to provide the average
flow value during the flow oscillations.
The average flow was obtained
from the average slope of the integral signal, recorded over a few
oscillation cycles.
The entire flow measuring instrumentation shown in
Fig. 2.4 performed quite satisfactorily at moderate flow oscillations.
However, when flow blockage, reversals or vapor flowback from the heated
sections occurred, large discrepancies were observed.
The venturi,
being non-symmetric, gave an erroneous reading for reverse flow even
without vapor entrainment.
As mentioned above, the large diameter ratio
venturi was chosen to minimize the pressure losses and was delivering a
very weak signal at low flow (0.04 psi at 1 ft/s).
It might have been
beneficial to use instead a simple orifice to get identical calibration
in both directions and a larger, direction-sensitive pressure signal
(a venturi gives a pressure signal of the same sign regardless of the
flow direction).
-
44
-
The response of the square-root amplifier was excellent, except in
the lower 5 percent of its full scale, where there were significant
deviations from the theoretical characteristic.
At these low velocities
the output was also sensitive to pressure signal drift, although the drift
was generally quite low (typically 0.001 psi).
to zero the transducer very carefully.
It was, however, necessary
For these reasons, the venturi
flowmeter was used mainly at high velocities and in the stable region.
At low velocities, below about 0.8 ft/s for oscillating flow during the
blowdown experiments, the flow was metered by volume as explained in 7.1.2.
The total flow to the test sections and the bypass was measured by
two orifices, connected to mercury manometers.
All flow measurements were
estimated to be accurate to a few percent.
2.2.3
Electric Power Measurements
The electric power supplied to each test length was monitored by a
set of seven panel wattmeters.
Provisions were also made for the
external connection of an accurate (Westinghouse, class 0.5%) laboratory
wattmeter whenever accurate measurements were desired.
The line voltage
was checked frequently during the experiments to detect and correct drifts.
2.2.4
Temperature Measurements
Copper-constantan thermocouples were used throughout the loop.
The
test-section thermocouples were steel sheathed, 0.062 in. in diameter.
Ordinary thermocouple wire in dry wells was used at other locations.
All
thermocouples were connected to a 144-point Honeywell-Brown potentiometric
recorder.
Switching of any thermocouple to a Visicorder channel for
dynamic recordings was also possible.
The recorder was calibrated
Mali,
-
45
-
frequently and was accurate to approximately 0.2*F.
2.2.5
Other Measurements
Occasionally, a hot wire probe, consisting of a 0.001 in. dia.
stainless steel wire, inserted through a thermocouple hole and stretched
across the flow area, was used as an inexpensive, average void fraction
gauge [46] to detect flow regimes and dynamic void fraction variations.
Its operation was quite successful and gave valuable qualitative results.
An attempt was also made to detect the dynamic variations of the
outside surface temperature of the test sections.
A measure of the
resistance variations of the conductive coating of the test sections by
a carrier frequency Wheatstone bridge failed for lack of sensitivity
(low temperature coefficient of resistivity) in
ment.
the 300 V power environ-
It would have been much better to use a resistance thermometer
(e.g. a thin insulated wire wrapped around the tube).
In any case, the
temperature variations were very small, of the order of 1 or 2*F even
during the most violent oscillations.
-
46
-
Chapter 3
PREDICTION OF THE STEADY-STATE OPERATING CONDITIONS
IN THE BOILING CHANNEL
The stability analysis that will be undertaken in Chapters 5, 6,
and 8 will require, like any stability analysis, knowledge of conditions
throughout the boiling channel.
Steady-state prediction methods
presented in this chapter will be experimentally verified in Chapter 4
in order to ascertain that the steady-state foundations of the stability
analysis are sound.
More specifically, a calculation procedure was developed in order
to predict the pressure, temperature, quality, and void fraction
profiles along the channel at steady state.
In essence, the whole
procedure centers around the pressure drop prediction; the additional
parameters are required inputs to the pressure drop calculation.
The
independent variables required are the inlet (all liquid) mass flow rate,
the inlet temperature, the exit (or condenser) pressure and the heat
flux distribution along the channel.
A computer code (PRESDROP) was
written in FORTRAN IV to carry out the calculations on an IBM-360 computer.
The code was written with the specific experimental geometry and fluid in
mind; however, it can be easily modified to accomodate any single channel
geometry and fluid.
In this code, turbulent flow at thermodynamic equili-
brium is assumed, the relative velocity between the phases is taken into
account, the fluid properties are variable with pressure, and the effects
of subcooled boiling are considered.
MM
-
11111W
-
47
An outline of the various correlations used in developing this
A description of the code and its
procedure is given in this chapter.
A parametric study of
FORTRAN IV listing can be found in Appendix A.
the pressure drop predicted under the present experimental conditions is
undertaken in the last section of this chapter.
3.1
General Calculation Procedure
The pressure drop along a single channel is calculated for a smooth,
constant diameter, round pipe and for vertical upward flow.
Reflecting
the experimental setup, the channel was subdivided into seven heated
lengths and a short unheated exit section, the heat inputs to each test
length being individually adjustable.
To further take into account in
the calculations the variations of the fluid properties with pressure
and the continuous variation of the quality, each test length was
subdivided into a number of segments, depending on the accuracy desired.
The input information is:
the inlet (all liquid) mass flow rate, w [lbm/hr]
the inlet temperature, T0 [F]
the exit or condenser pressure, pe
the heat input distribution, q
[psia] or pcond [in. Hg]
, i = 1,7 ordered from inlet
to exit [Watts per test length]
The total net heat input is then given by
7
qtnq
=
.Z
qi --qi
q. loss )[Btu/hr]
(q.
[t/r
where the losses are estimated from a previous iteration, Section 3.7.
The inlet enthalpy and the total enthalpy rise are given by
h. = h (T ) ,
f o
in
Ah = q
/w
tn
[Btu/lbm]
-
48
-
The exit quality can then be calculated as follows:
h.
h in
ex
+±Ah-h(p
f
h fg(p
ex
)
)
The conditions now being known at the exit, the pressure drop is
calculated for each segment, marching upstream until the channel inlet is
reached.
The two possible cases of subcooled and two-phase flow as well
as the special case of the segment around the bulk boiling boundary are
treated in the following sections.
All properties are evaluated at the
saturation pressure or temperature unless otherwise specified.
3.2
Pressure Drop
The pressure drop at steady state is given by the sum of the
frictional, gravitational, and acceleration pressure drop terms:
Ap = Apfr
gr
ac
It becomes necessary at this point to define rigorously a few terms
that will be used extensively throughout this work.
The (bulk) boiling
boundary (BB) is defined as the point in the channel where the bulk or
mixed-mean enthalpy of the flow reaches saturation.
The subcooled region
extends from the inlet to the BB, and will occasionally be referred to as
single-phase region (subscript 1),
boiling.
(NVG)
ignoring the occurrence of subcooled
The boiling region starts at the point of net vapor generation
and is
divided by the BB into a subcooled boiling and a (bulk)
boiling region,
otherwise referred to as two-phase region (subscript 2),
whenever the presence of vapor in
the subcooled region is
superheated vapor region is not considered in
this work.
ignored.
The
INCH~.
-
49
-
Frictional Pressure Drop
3.2.1
3.2.1.1
Single-Phase Region
For a single-phase diabatic flow, according to Sieder and Tate [47],
Ap
4
lfr
where
f = f(Reb)
b
and
AL
D
Re
2
G
2p
0.14
GD
p
The friction factor for smooth pipes is
Reynolds number
the wall is
formulated as a function of the
for use in the program (Appendix C).
The temperature of
obtained using the familiar McAdams correlation for the heat
transfer coefficient
k
= 0.023 (Reb) 0.8 (Prb0.4
There might be some question concerning the validity of the wall-tobulk temperature correction when the wall temperature exceeds saturation
temperature.
Radial property variations might also be expected to
influence the heat transfer coefficient; however, this was not considered
since some uncertainty in
the wall temperature could be tolerated.
With
this procedure, the predicted pressure drop slope agreed well with the
observed one in the single-phase region, even under extreme wall temperature conditions (Figs. 4.4 and 4.5).
Finally note that, lacking an
adequate correlation for Freon, the effect of subcooled boiling on the
frictional pressure drop is ignored.
-
3.2.1.2
50 -
Two-Phase Region
A number of more-or-less sophisticated empirical and semi-empirical
models exist for the calculation of the frictional pressure drop in the
two-phase region (e.g. [49,50,51,52]).
The Lockhart-Martinelli model [49]
was used here because of its wide acceptance and the similarity of the
experiments from which it
was derived to the present one.
The calculation
outlined below:
procedure is
2tt
Afr 2
2
AL
D
S-4
[G(l - x)]
2
pf
f = f(ReZ )
Re
G(l -x) D
if
The Lockhart-Martinelli two-phase friction multiplier
2
tt=
ktt
2
@Ztt (Xtt)
is obtained numerically from a polynomial which was fitted to the
recommended curve as a function of the parameter
Xtt
f 0.1
1 -x\0.9 (g)0.5
tt
See Appendix C.
ptt
1
Both phases are assumed to be in the turbulent regime,
which is not strictly correct for the gas phase near zero quality and for
the liquid phase at low flow rates and high qualities.
Practically,
however, it does not make any difference as these regions cover only a
short channel length.
In the above formulas, the evaluation of the
- _11HU,
-
51
-
quantities at the average quality for each segment
x =
2 (xe +x.)i
requires knowledge of the segment inlet quality x., which is obtained by
iteration.
Gravitational Pressure Drop
3.2.2
This term is
Apgr
given simply by
[f(l -
) + p
a] g AL
The difficulty resides in the estimation of the average void fraction,
in the boiling region.
a =
a,
The Lockhart-Martinelli void fraction correlation
a(X tt)
was used throughout this workwhere again the empirical function was
approximated numerically by a polynomial (Appendix C) and all quantities
were evaluated at the average quality
x
The Lockhart-Martinelli void fraction correlation was, however,
compared to other correlations as discussed in Section 3.4.
Section 3.5
describes the evaluation of the true quality in the subcooled boiling
region, which can be used with the Lockhart-Martinelli correlation to
estimate the subcooled voids.
The expression reduces to the usual
hydrostatic expression in the region prior to subcooled boiling.
3.2.3
Acceleration Pressure Drop
Assuming that each phase is flowing separately with uniform velocity
throughout the flow cross-section, the acceleration pressure drop is
given by [53]
-
Ap
ac
=
B(x
e1
e,p)
-
52 -
B(x.,p.)
i
with
B(x,p)
-G 2
(1
x
- x( 2
a)p
ap
-
fg
In
the single-phase region, B(x,p) reduces to
B(T)
-G
Pf
3.3
The Segment Containing the Bulk Boiling Boundary
It is noted that since there is a large pressure drop at the bulk
boiling boundary due to the sudden acceleration of the fluid, it becomes
necessary to treat separately the segment that contains the zero quality
point.
This effect is especially pronounced at low pressures where the
equilibrium void fraction rises from zero to 50 or 75 percent in a few
inches of heated length.
It is then necessary to determine the zero quality point by successive
iterations, calculate separately the pressure drops in the non-boiling and
boiling lengths, and add them to get the total pressure drop.
Although
the solution to the problem seems trivial, considerable numerical difficulties were encountered.
The difficulties are significantly alleviated
when subcooled boiling is taken into account, due to smoothing of the void
fraction distribution around the bulk boiling point.
this problem in more detail.
Appendix A discusses
__11IN11,111111
-
53
-
Prediction of the Void Fraction
3.4
Among the numerous existing void fraction correlations, four were
chosen to be compared here:
a.)
The Lockhart-Martinelli empirical correlation [49]
a =
a(X tt)
with
Xtt
Xtt (xp)
neglects the effects of mass flux, pipe diameter, and flow regime but has
given satisfactory results for a variety of fluids and conditions.
b.)
A correlation proposed by Zuber and Findlay [54] for the "churn-
turbulent" flow regime, which takes into account the flow regime and the
mass flux effect, is given by
Xa =
p
c.)
1.13[
pP9
+ 1 -x]
Pf
+
1.18
G
g(Pf P
P2
Pf
1/4
The slug flow model proposed by Griffith and Wallis [55], which
takes into account tube diameter, flow regime and mass flux,
Q /A
Q + Qf
1.2(
with
wx
g
d.)
AA
+ 0.35(gD)
and
P
Q
f
1/2
w(l - x)
Pf
The simple homogeneous model given by
x
x
+ (1 - x)Pg /Pf
-
54
-
These four models were compared at typical experimental conditions
of this investigation as shown in Fig. 3.1.
The dotted lines indicate
that the model is probably used outside its intended range of flow regime.
The Lockhart-Martinelli extension towards the low quality region is an
arbitrary one, as noted in Appendix C.
range of interest, i.e. for
The comparison shows that in the
x = 0.01 to 0.30, the first three models
agree closely, while the homogeneous model overestimates the void fraction.
It was felt that this comparison justified use of the Lockhart-Martinelli
void fraction correlation throughout this work.
The effect of variation
in the predicted void fraction on the pressure drop is discussed in
Section 4.5.
3.5
Void Fraction in
the Subcooled Boiling Region
The void profile in
the subcooled region is
by the position of the point of NVG.
essentially determined
This is considered to be the point
in the channel where the void fraction increases rapidly with position.
Upstream of NVG, the subcooled voids are on the wall and two-phase effects
are small.
Levy [56] and Staub [57] have somewhat similar approaches to
prediction of this point.
They consider NVG to be coincident with
bubble departure which is due to unbalance of forces on the bubble.
Staub
takes into account surface tension, buoyancy and shear drag terms, while
Levy considered only surface tension and buoyancy effects.
Both models
have "adjustable constants" which have been evaluated mainly from high
pressure water data, and there is
no assurance that they should apply
equally well to Freon-113 at low pressures.
The two models were compared for the present experimental
conditions
W
o
-
55
-
using the following recommended values of the constants.
Staub:
f(0) = 0.030
Levy:
C = 0.015
and
C' = 0
The subcooling at the point of NVG, as predicted by the two models, is
plotted in Fig. 3.2 versus the mass flux, for various heat fluxes.
As
expected, Levy's model gives unrealistic estimates at very low velocities
where the buoyancy effects dominate or at least play an important role,
but the models agree otherwise.
f(3) = 0.030 was used
Staub's model with
in most of this work, although some earlier calculations were done using
Levy's model.
Once the point of NVG is known (given by the models as subcooling
with respect to the local saturation temperature
postulated between the "true" local vapor quality
ponding equilibrium value
x tr=x-x
x.
xtr
and the corres-
Levy proposes:
exp(x/x d - 1)
for x > xd
for x < xd
= 0
x
ATd) a relation is
where:
c ATd
x
d
=
-
hfg
< 0
The void fraction is then calculated from the appropriate relationship
(in the present case the Lockhart-Martinelli correlation) using this true
quality.
The computer subroutines (SUBBOL) that perform these calculations are
listed in Appendix A.
on the pressure drop is
The effect of the subcooled boiling void fraction
examined in
Section 4.4.
-
3.6
56
-
Temperature Profile
In the subcooled region, the local bulk temperature is directly
related to the fluid enthalpy.
A correction is, however, necessary to
account for subcooled boiling:
Tb= T [h -
x trh
(p)]
In the bulk boiling region the local temperature is equal to the
saturation temperature.
3.7
Heat Losses
The glass channel was not insulated in
observations
order that continuous visual
of the boiling phenomena could be made.
The heat loss from
the outer wall was small, of the order of 5 percent, but not negligible.
Gouse and
iwang [27] under almost identical experimental conditions have
measured these losses and tested the accuracy of the correlation used to
predict them.
The good agreemenL they obtained justified the use of
similar methods to predict the heat losses here.
The film temperature drop is calculated, neglecting the heat loss
as a first approximation.
The McAdams correlation is used in the forced
convection region and the heat transfer coefficient is set equal to
1000 Btu/hrft *F in the boiling region.
Then the wall temperature drop
is estimated for an average glass temperature.
These two temperature
drops added to the average bulk temperature yield the average outside wall
temperature, Tout.
q
nc
The natural convection losses are then given by [58]
= 0.29 AT
a
/L4
[Btu/hr ft ]
mill,
-
57
-
with
a
[*F]
- T
= T
AT
room
out
L = test length
[ft]
Adding the radiation loss, assuming a view factor of 1.0 and emissivity
of 0.95, the total heat loss is obtained.
The FORTRAN listing of the
subroutine LOSSES in Appendix A provides additional details.
3.8
Parametric Study of the Pressure Drop
As all the stability predictions depend closely on the steady-state
pressure and quality distributions at the threshold of stability and on
the relative magnitude of the three terms in the pressure drop equation,
it will be useful to examine here the parametric effects of mass flow
rate, inlet temperature, heat input, and quality under the present experimental conditions.
3.8.1
Effects of the Inlet Temperature and the Mass Flow Rate
Figure 3.3 shows the variations of the total pressure drop, Ap1 ex,
when the inlet temperature is
changed, while all the other independent
variables are maintained constant.
mass flow rate.
The subcooled boiling and the heat losses were not taken
into account in these calculations.
pbb,
Figure 3.4 shows the effect of the
the exit quality,
x
The pressure at the boiling boundary,
, the length of the single-phase region,
and the subcooling with respect to the boiling boundary, AT subb
T
are also shown in these figures,
stability analyses.
as they all
=
zbb'
Tsat ('bb
have some importance in
the
A detailed discussion of these parametric trends is not
-
58
-
required here since the explanation for the behavior is
quite simple and
the plots will be considered later.
Relative Importance of Friction, Gravity, and Acceleration Terms
3.8.2
For a given geometry, fluid, and pressure level, the local differential
frictional and gravitational pressure drops are functions of the mass flux
and the quality only.
plane.
It is thus possible to compare them in the (G,x)
The acceleration term depends on the rate of vapor generation,
or the rate of quality increase with length and therefore will also be
dependent on the heat input.
The three terms were calculated for a matrix of values in the (G,x)
plane and for three different heat fluxes.
by computer for lines of constant ratios.
Contour maps were than produced
Figure 3.5 shows such a contour
The plot has a background of lines of equal gravity-to-friction
map.
ratio.
Superimposed on these are the three unity ratio curves: gravity
friction = acceleration (for q' = 792 Btu/hrft, 300 W/test length).
subdivide the entire plane into three major regions.
=
They
Gravity is dominant
in the upper left region, acceleration is larger than the two other terms
in the lower left corner and friction is the major term in the rest of the
plane.
The 264 and 1320 Btu/hrft (100 and 500 W/test length) lines,
also
shown on this plot, permit to evaluate the effect of the heat flux on the
acceleration term: as the heat flux is increased, the "triple point" moves
upwards along the gravity = friction line, enlarging the acceleration
dominated area.
chapters.
This figure will be referred to frequently in subsequent
wmwwww
1116
-
59
-
Chapter 4
MEASUREMENT OF STEADY-STATE PRESSURE AND TEMPERATURE PROFILES
AND COMPARISON WITH PREDICTIONS
Detailed pressure and temperature profile measurements at steady-state
were undertaken in order to verify the validity of the computational
methods developed in the preceding chapter.
The range covered in the 21
experimental runs was:
inlet temperature, T
95 and 115 *F
gross average heat flux, q"
0, 4700, 9400 and 14,100 Btu/hrft 2
(0,
200, 400 and 600 W/test length)
inlet velocity,
V
approximately 0.5 to 10 ft/s
condenser pressure, pcond
atmospheric
heat flux distribution
uniform, cosine,
and "rooftop"
Heat flux distributions are given in Table 4.1.
Table 4.1
Definition of the Heat Flux Distributions Used Throughout This Work
Local to Average Heat Inputs (per Test Length)
1
2
3
4
5
6
7
1
1
1
1
1
1
1
Chopped Cosine
0.43
0.98
"Rooftop"
0.875 1.75
Test Length Number
Uniform
Distribution
1.35
1.48
1.35
0.98
0.43
1.458 1.165 0.875 0.585 0.291
-
60
-
The experimental procedure is outlined in the following section.
The
effect of the various uncertainties in the theoretical models on the
calculated pressure drop and the effect of the heat flux distribution are
also discussed in
4.1
this chapter.
Experimental Procedure
All experiments were conducted with a single channel (Channel C).
The
other two channels were isolated during the measurements using valves
Vll
and V12 (see Fig. 2.1).
After slowly heating up the system, the Freon at an inlet temperature
close to saturation was circulated in all three channels at full power
for approximately one hour to extract dissolved air.
During this period
the main condenser water was shut off and most of the condensation occurred
in the small after-condenser which was left permanently open to the
atmosphere during these experiments.
purge of the air from the system.
This technique resulted in a partial
Then the inlet temperature was set by
adjusting the steam and water flows to heat exchangers
HXl and HX2.
The
pressure transducer lines were carefully purged and the transducers
electrically zeroed at the nominal run temperature with a negligibly small
flow in the test section, which was necessary to maintain a constant head
and to purge any bubbles.
Valve
flow was controlled using valves
V9 (bypass),
V13
was then completely opened and the
V10 (common
and Vl (pump outlet).
inlet to the test sections),
Since some of the data were taken in
regions which would normally be unstable, the oscillations were damped out
by using inlet throttling provided by valve
V10.
It was generally
necessary to supply some cooling to the main condenser during the measure-
-
-
61
ments in order to achieve a constant channel exit pressure.
The experiments began at a flow rate as low as the stability of the
system would permit, then the flow was increased in small increments.
Flooding of the riser, which had an insufficient liquid drain, defined the
maximum flow rate.
For each data point, the inlet flow, w, the inlet pressure, p0, the
pressure at stations
1 to 8,
p1 to p8 ,
the inlet temperature, T ,
occasionally the temperature at the remaining stations,
recorded.
and
T2 to T8, were
The instruments were checked frequently for drift, and correc-
tions to the data were made whenever necessary as described in Chapter 2.
As preliminary measurements with isothermal single-phase flow had
shown that the thermocouples were disturbing the flow and the pressure
measurements
(probably by creating turbulence at the diametrically
opposed pressure taps), they were retracted flush to the wall for most of
the steady-state experiments.
4.2
Measured Pressure and Temperature Profiles
Typical data are presented and compared to the theoretical predictions
in Figs. 4.1 - 4.7.
To make the local slope changes in the pressure
profile visible, the hydrostatic, all liquid, pressure head at nominal
run temperature was substracted from the absolute pressure.
The plotted
value therefore corresponds exactly to the pressure transducer signal.
Appendix D gives a summary of the experimental runs and a tabulation of
the data.
Figures 4.1 to 4.3 show the dependence of the total pressure drop
across the heated section,
p1 - p
,
on the mass flow rate, for two
-
62
-
different inlet temperatures at three power levels.
Both the uniform and
the chopped cosine heat flux distribution results are plotted in these
figures.
In one instance, the "rooftop" distribution is also included.
Figures 4.4 to 4.6 are randomly selected pressure profiles along the
channel.
4.2.1
Total Pressure Drop at the Threshold of Stability
The accurate prediction of the steady-state pressure drop at the
threshold of stability and at other dynamically interesting points is of
paramount importance in this work.
A final test of the calculation methods
developed in the preceding chapter is made here by comparing the total
pressure drops
(p1 - pex ),
measured during the stability experiments
described in the following chapter, to PRESDR predictions.
This comparison
given in Fig. 4.7 includes all threshold and transition points of runs
D4 to D21.
For the PRESDR calculations the channel was subdivided into
14 segments, the heat losses and subcooled boiling according to Levy's
model were taken into account, and one major iteration was required for
each experimental point.
4.3
Accuracy of the Predictions
The accuracy of the steady-state pressure drop predictions was
generally good to excellent.
Some discrepancies that were noted are
analyzed below.
4.3.1
Pressure Drop Predictions
Table 4.2 gives the average predicted to measured total pressure drop
ratios and associated rms errors for all the points plotted in Fig. 4.7.
-
63
-
There is apparently a systematic but small error in the predictions as
the average ratio is 0.97.
The
6%
rms error is otherwise quite
satisfactory.
Table 4.2
Accuracy of the Total Pressure Drop Predictions
at the Threshold of Stability and the Transition Points
Ave. predicted
to measured ratio
rms
error
Heat Flux
Distribution
Number
of points
D4-D23
uniform
77
0.974
0.062
D4-D23
cosine
25
0.970
0.053
TR, E
uniform
23
0.955
0.066
E
unheated inlet
12
0.985
0.025
137
0.971
0.059
Runs
All
-
See also Fig. 4.7
Inspection of Figs. 4.1 to 4.3 indicates that the agreement between
the measured and predicted total pressure drops is similar to that
exhibited by the data of Fig. 4.7.
The only noticeable disagreements are
in the low exit quality, high velocity region (x = 0.01 to 0.05) where the
experimental curves show an accentuated hump.
The pressure profiles plotted in Figs. 4.4 to 4.6 show invariably a
disagreement between the predicted and measured pressure increment near
the channel exit at low exit qualities and sufficiently high mass fluxes.
The experimental profiles do not exhibit the predicted sharp pressure
drop (caused by the acceleration term at the BB or the point of NVG),
suggesting thermal non-equilibrium due to poor nucleation.
Thermal non-
-
64 -
equilibrium could also arise from an insufficient rate of vapor generation
when the flow is leaving the channel before reaching thermal equilibrium.
In fact, as the friction and acceleration terms are dominant in this
region (Fig. 3.5), a lower true exit quality, resulting from either
mechanism, would produce a significantly smaller pressure drop at the
exit, shifting the entire pressure profile downwards.
The same trend is
evident in Fig. 4.7 where the predicted-to-measured pressure drop ratio
is generally larger than one at small exit qualities.
At higher qualities
and lower mass fluxes the flow is given sufficient time in the channel to
reach equilibrium and these effects disappear.
The effects of poor
nucleation were dramatically exhibited in a hysteresis phenomenon that
was observed and is reported in Section 4.6.
Figure 4.6 shows an opposite effect.
PRESDROP exit pressure drop
predictions are low for all but the highest and the lowest mass fluxes.
Three different and opposing effects must be considered here:
a.)
The separated flow model used here systematically underestimates the very important acceleration pressure drop in this
region
b.)
(page 74 of [59]; Table 4.3).
The almost uncontrollable air content of the Freon has an
important effect on the void fraction, as a recent investigation
has shown [60].
c.)
Void fraction measurements with Freon-22, reported by Zuber et
al. [61] reveal a discrepancy of the Lockhart-Martinelli
correlation, which was derived from adiabatic flow data and,
moreover, does not account for the mass flux effect.
The
measured void fractions were always below the Lockhart-Martinelli
predictions, the error being largest for the lowest mass fluxes.
, I''11,
1,jIII11,
1110110510
-
65
-
It appears that in Fig. 4.6 the first two effects (especially
the air content of the Freon) were overwhelming.
Excluding
the high mass flux points (where poor nucleation at the exit
is evident), the measured exit pressure drops are higher for
all points except at 0.92 ft/s.
Inspection of Fig. 3.5 will
confirm that only this point is in the gravity dominated
region.
There are almost no discrepancies of this nature in
Figs. 4.4 and 4.5.
4.3.2
Temperature Predictions
Acceptable agreement between the measured and the predicted
temperature profiles was obtained, considering that no particular care
was taken to minimize the thermocouple errors in this investigation.
The hypothesis of a deficient vapor formation near the exit at
high mass flow rates, made in the previous section, seemed at first to
be in disagreement with the temperature readings at station 8 which
were lower than the saturation temperature, suggesting excess vapor
formation.
It was found however that this discrepancy was entirely
due to a thermocouple error.
In contrast to the other test stations
which were made of plexiglass, station 8 was made of a massive brass
block having a much larger heat capacity and conductivity, and rather
large diameter insulated thermocouples were used.
temperature reading at station
8
This resulted in a
that was consistantly below
-
66
-
the saturation temperature by as much as 10 to 15 *F.
When, however, the
0.062 in. OD insulated thermocouple was replaced by a 0.001 in. bare
thermocouple, the discrepancy disappeared and temperatures slightly in
excess or equal to the saturation temperature were recorded.
4.4
Effect of Subcooled 3oiling
Visual observations have shown that, at low flow rates, detached
bubbles, approximately 1/32 to 1/16 in. in diameter, were produced from
discrete nucleation centers of the heated wall, only a few inches from
the inlet, regardless of the heat flux intensity.
At higher velocities,
approximately above 1 or 1.5 ft/s, the first nucleation centers were
generally pushed downstream up to an experimental station where numerous
small geometry defects provided good nucleation centers.
Both Levy's
and Staub's predictions of the point of net vapor generation (see 3.5)
were always well downstream of the visually observed values and there was
no way to fit the data by any reasonable change of the model constants.
Despite these facts, the pressure drop predictions in the subcooled region
were good, suggesting that the opposite effects of the subcooled voidage
on the gravity and frictional terms probably balance.
Figures 4.4 and 4.6 show the effect of the two models on the
calculated pressure profile.
Again, the differences are small and confined
to a narrow region extending from the point of net vapor generation to the
bulk boiling boundary.
4.5
Choice of the Void Fraction Correlation
As shown in Fig. 3.1 there is little difference in the predictions of
NFONN
-
67
IIINNIWI
-
the three void fraction correlations that do take into account the
relative velocity between the phases.
To estimate the effect of the
slip on the pressure drop, values calculated with the Lockhart-Martinelli
void fraction correlation (L-M) are compared here to predictions made
using the homogeneous void fraction model.
The respective void fractions
were used to evaluate the acceleration and gravitational pressure drop
formulas of Sections 3.2.2 and 3.2.3.
The frictional pressure drop term
was calculated in both cases using the L-M method.
A few representative points at the threshold of stability were used
for this comparison and the results are summarized in Table 4.3.
The
reported pressure drops are as usual measured from the entrance of the
heated section to the channel exit (p1
p
).
For the few points tested here, the predicted to measured pressure
drop ratio averaged 0.932 for the homogeneous and 0.992 for the L-M
correlation.
The deviations from the average, as characterized by the rms
error were 9.2 percent for the homogeneous and only 5.7 percent for the
L-M method.
4.6
Observed Hysteresis in the Pressure Drop Characteristic
As described in 4.1, the pressure-drop - flow-rate characteristics
were generated by gradually increasing the flow rate.
When, however, at
the end of a run, the fla was reduced again to repeat a few points as
a check, it was observed that very often these points would not lie on the
increasing flow characteristic.
The explanation that this behavior was
due to suppression of the nucleation at high mass flow rates was advanced
to explain the discrepancies.
When the flow is reduced there is a
Table 4.3
Effect of the Relative Velocity Between the Phases (As Predicted
by the L-M Correlation) on the Gravitational and Acceleration
Pressure Drop
qw.
T
w
x
Lockhart-Martinelli*
gr
lb
[W/TL]
[*F]
[
m]
ac
fr
Ap
Ap
Ap
Ap
Ap
Ap
Point
Homogeneous Model*
gr
ac
fr
pressure drop
ratio
L -
[psi]
[psi]
Calculated to
measured total
M
Homog.
D20-194
500
41.7 396 0.196
4.344 0.352
O.9U8
4.088 0.774 0.900
0.978
1.006
D20-181
400
43.0
192 0.496
2.899 0.358 0.937
2.539 0.526 0.942
0.922
U.881
D19-191
300
40.0 234 0.203
4.227 0.132 0.412
3.914 0.290 U.416
0.992
U.961
D19-986
300
42.0
119 0.66.
2.37u 0.237 u.645
Z.17 U.291
U.49
1.1/1
1.U22
D4 -299
300
120.3
387 0.293
1.849 0.06d
2.415
1.314 1.1b
t.408
U.961
0.964
D4 -917
200
118.0
281 0.260
1.846
0.267
1.311
1.214 u.537
1.331
0.946
U0.85
D17-296
100
85.3 1,54 0.117
4.103
0.028 0.140
U.068 0.145
l.U24
U.935
D4 -294
1U
1.0u6
u.846
*
116.6
286 0.120
2.759 u.l0U
0.622
3.687
2.034 v.241
U.653
Small differences in the friction terms are due to change of properties with pressure
I''
-
69 -
substantial delay before normal nucleation resumes.
The main effect should
be localized towards the channel exit.
The suspected hysteresis was revealed by making careful measurements
at both increasing and decreasing flow rates, together with bulk temperature
measurements.
Figure 4.8 shows the measured pressure characteristics at
stations 1 and 6.
The effect is obviously at the exit, as the station 6
characteristic shows.
The measured temperature profiles for two points
lying on the two distinct branches of the curve are plotted in Fig. 4.9
which clearly shows that, on the decreasing flow rate curve, the flow was
superheated toward the channel exit (the drop in temperature at the exit
is a thermocouple error as pointed in 4.3.2).
Figure 4.8 also shows that
it was necessary to go above a certain velocity to induce the hysteresis
cycle.
Note that it was practically impossible to obtain any data on the
negative slope part of the decreasing-flow branch: The return to normal
nucleation was abrupt.
There was, however, also a very slow time recovery
that would take place in several minutes; points on the lower part of the
curve would very slowly drift towards the upper curve.
A similar but much more violent phenomenon was observed during a
later experiment in which the flow in the unheated channel was at a
temperature close to saturation.
There was absolutely no voidage in the
test sections until the entire channel voided suddenly, in less than a
second.
Such sudden flashing can be explained by the strong dependence
of the saturation temperature on pressure at the low pressures of these
experiments.
Apparently there was an avalanche effect, the voided upper
parts of the channel reducing the pressure at lower points and creating
further voids.
-
70
-
The occurrence of such nucleation instabilities should be kept in
mind when examining other types of flow oscillations as they might trigger
and enhance them.
4.7
Unfortunately, they are practically unpredictable.
Effect of the Heat Flux Distribution on the Pressure Drop
The effect of the heat flux distribution will be discussed now by
examining one-by-one the three terms of the pressure drop equation.
The acceleration pressure drop depends only on the inlet and exit
qualities and therefore is
completely independent of the heat flux
distribution within the channel.
The gravity contribution comes mostly from the single-phase region.
Any distribution that would result in downstream shift of the boiling
boundary should therefore increase the gravity head.
The frictional pressure drop term becomes dominant near the exit
of the channel, where the flow attains large velocities.
In most cases,
all the frictional pressure drop is concentrated near the exit.
follows that a downstream movement
It
of the boiling boundary will reduce
the frictional pressure drop slightly as a two-phase portion of the channel will be replaced by a single-phase portion which has a smaller friction
contribution.
The net effect will of course depend on the relative impor-
tance of gravity and friction (Fig. 3.5), but it is obvious that for
relatively smooth distributions the change will be small.
The calculated
and experimental results of Figs. 4.1 and 4.2 confirm this observation.
4.8
The Pressure Drop in the Unheated Entrance Piping
An accurate estimate of the pressure drop between stations 0 and 1
-
71 -
(including the flow venturis and the valves) will become necessary for
Using all the available
the dynamic predictions, later in this work.
data from the steady-state runs the following correlation was obtained:
Ap
=
(1.5 ft) p/144
=
0.00177
=
7.01 x 10
(4.1)
01gr
Ap
p0
8
0.2
V1.8
Olfr
-10
1.8
G
y
0.2
(4.2)
/p
where the pressure drop is in psi, p in lbm/ft3, y in lbm/hrft,
and G in lbm/hrft
V in ft/s
2
The frictional pressure drop from the bypass to the inlet of the
test sections through the preheater (HX2) is negligible.
4.9
Conclusions
The data presented in this chapter show that an accurate prediction
of the pressure drop in low pressure systems is possible when well established correlations are used, the channel is subdivided into a relatively
small number of segments, and the variations of the fluid properties with
pressure are taken into account.
The predictions were good in a
relatively wide range of variables and particularly good (average error
3%, rms error 6%) in the region of interest for dynamic tests (for exit
qualities mostly around 30 percent).
The only discrepancies were due to
unpredictable deviations from thermodynamic equilibrium.
Subcooled
boiling was shown to have only a small local effect on the pressure drop.
The Lockhart-Martinelli void fraction correlation gave systematically
better results than a void fraction based on the homogeneous model.
-
72
-
The heat flux distribution was shown to have only a minor effect on the
total pressure drop.
Given the good agreement at steady-state it is felt that the
dynamic calculations could be tackled with some confidence.
-
73 -
Chapter 5
DYNAMICS OF THE SINGLE-PHASE REGION
The single-phase region is
defined here as the channel length
extending from the inlet plenum (station 0)
to the boiling boundary (BB),
i.e. to the point where the mixed mean enthalpy is at saturation.
It
includes the unheated upstream portion of the channel between stations
0 and 1.
Any occurrence of subcooled boiling in this region is
neglected in
vary in
this chapter.
The length of the single-phase region will
time when the flow oscillates.
The linearized coolant energy equation,governing the time variation
of enthalpy in the single-phase region under oscillating flow conditions,
will be coupled with the transient conduction equation of the heated
wall to yield the flow-to-local-enthalpy transfer function,
any axial position.
H(z,jw), at
The two sets of equations mentioned above will be
linked by a flow-dependent heat transfer coefficient.
Although the heat
input to the wall will be maintained constant, there will be time
varying heat input to the coolant because of heat storage in
the wall.
A number of investigators [62,63,65,66,67] have considered the
problem of heated wall dynamics.
The incentive was generally to predict
the thermal behavior of the fuel in a nuclear reactor under transient
conditions, and especially during oscillator tests when some variable,
generally the power, is externally oscillated in order to measure the
dynamic characteristics of the reactor.
vary in
complexity in
The transfer functions obtained
inverse proportion to the number of simplifying
-
74 -
assumptions made in their derivation.
In the present study, almost all
simplifying assumptions were avoided.
The solution is identical to the
one obtained by Smets [63] for the case of oscillating flow.
simpler derivation is
However, a
presented here; the resulting transfer function
is
put into a much more compact and meaningful form, and the solution is
specialized to represent the exact experimental conditions.
Once the wall temperature and the enthalpy variations along the
channel have been determined, it is relatively easy to deduce the timevarying position of the boiling boundary, again using a linearized model
for simplicity.
In the present work, some effects that had been neglected
in previous analyses, because they were relatively unimportant in high
pressure systems, are considered and shown to be of some importance in low
pressure systems.
These are the pressure variations at the instantaneous
position of the boiling boundary due to the gravity head, to the frictional
losses in the single-phase region, and to the inertia of the subcooled
liquid column.
Finally, knowing the instantaneous position of the boiling
boundary and the instantaneous flow, the pressure drop variations in the
single-phase region will be obtained in
a straightforward manner.
In this chapter, all the formulations will be treated in a completely
linearized form.
linear.
In fact, the wall conduction equation is inherently
An exact non-linear solution for the coolant energy equation can
be obtained numerically when a steady and uniform heat input to the
coolant is assumed (Section 5.2.4.1).
This solution was not used, however,
in order to save computation time and to facilitate the incorporation of
the wall dynamics into the model.
The comparison with the exact solution
will confirm, as expected, the validity of the linear solution at the small
-
75
-
relative amplitudes, which occur at the threshold of stability.
The linearized equations will be Laplace-transformed in time and
then integrated in space.
Although, in the sections that follow, the
Laplace vaxiable, s, and its imag:nary part, jw, the imaginary angular
frequency, are used interchangeably, it is understood that the transfer
functions will be evaluated along the imaginary frequency axis
jW.
Subroutine DZDWTF was written in FORTRAN IV to perform the numerical
computations on the IBN-360 and is listed in Appendix F.
5.1
Dynamics of the Heated Wall
The transient conduction equation of the heated wall will be solved
in this section.
It is recalled that heat is supplied to the test sections
by means of the electrically conductive external coating.
The following
assumptions are made:
a.)
The heat generation rate is
constant.Since the temperature
coefficient of resistivity of the oxide coating is
of the order of 10
4/ 0 C,
this assumption is fully justified.
b.)
The coating can store no significant amount of energy.
reasonable for the 16 pin.
c.)
coating.
The heat losses are either negligible or time independent.
Section 3.7 it
d.)
This is
In
was noted that the losses are less than 5 percent.
The thin cylindrical glass tube wall is
of equal thickness.
approximated by a plate
Although the solution for a cylindrical geometry can
be obtained at small additional complication
(namely,
replacement of the
hyperbolic functions that will appear in the solution by the corresponding
-
76
-
Bessel functions), the plane solution was retained to simplify the
numerical work.
e.)
The axial heat conduction in the wall is neglected.
f.)
Average values of the thermophysical properties of the glass
wall are used.
g.)
The heat generation and the heat transfer to the coolant are
circumferentially uniform (one dimensional problem).
h.)
The heat transfe:- at the wall-fluid interface will vary
according to the instantaneous value of the flow.
i.)
Any axial heat flux distribution is
permitted.
Z
w
Tb(zt)
q"(z)
0
---
q"(z,t)
d
SX
FIG. 5.1 THE HEATED WALL
The conduction equation at axial position
in
Fig.
z,
for the geometry shown
5.1 becomes
2
2T(z,xt)
X2
1
a
3T(z,x,t)
t
(5.1)
millimilism
-
where
mill
77 -
is the thermal diffusivity of the wall.
a = (k/pc)w
In accordance
the boundary condition at the heated side is
with assumptions a.) to c.)
q"(z)
3T(z,x,t)
3x
_
where
(5.2)
x-Ok
x=0k
is the applied constant heat flux.
q"(z)
0
At the cooled side, the
boundary condition is
h (t)
-T(z,x,t)
DX
hc
[T(z,dt)
- T (zt)
b
k
x= d
where T(z,d,t) is the surface temperature at axial position
the corresponding fluid bulk temperature, and
hc (t)
(5.3)
z, Tb(z,t)
the time-dependent
(flow dependent) convective heat transfer coefficient.
It is convenient to write
T(z,x,t)
Tb(z,t)
T'b(z) +
=
(5.4)
T*(z,x) + 6T(z,x,t)
=
6
(5.5)
Tb(z,t)
(5.6)
h (t) = h* + Sh (t)
c
c
c
where the superscript
denotes the steady-state conditions satisfying
0
the time-independent equations
2 T*(z,x)
0
(5.7)
ax
q"(z)
0
and the
6
= h*
c
[T*(z,d)
- T
(z)] =
-
k aT(zx)
(5.8)
baxd
denotes small perturbations from equilibrium.
As stated in assumption
h.)
above, if the heat transfer coefficient
-
78 -
can be expressed as
constant *wa
h*o=
0
C
for small flow variations,
6hc
-
a 6w(t)
~
ho
(5.9)
w0
c
Lacking a better discription of the transient convective boundary condition, it is hoped that this formulation will be valid, at least at low
frequencies
[69,70].
Inserting definitions (5.4)
to (5.6)
into Eqs. (5.1)
eliminating the stEady-state terms from these equations,
Eq.
the only second-order te-rm in
36T(z,x,t)
a2
_
_1
oL
a6T(z,x,t)
_6T(z,x,t)
x
(5.8)
and neglecting
yields
36T(z,x,t)
9t
=
(5.10)
0
C [6T(z,d,t)-
xd
to (5.3),
d
(5.11)
6T (zt)]
b
+ E(z)
5w(t)
(5.12)
ho
where
and
C
(5.13)
=
a
k
q"(z)
w
*F/ft]
lbm/hr
Notice that in the equations written above the axial coordinate
(5.14)
z
is
merely a parameter introduced through the boundary condition, in particular
Tb'
-
79
-
Eqs. (5.10) to (5.12) will be now Laplace-transformed in time*,
.;ith the initial condition
2 6T(z,x,s)
6T(z,x,0) = 0:
(5.15)
s 6T(z,xs)
2
3x
a6T(z,x,s)
ax
(5.16)
=0
_6T(z,x,s)
d [6T(z,d,s) -
ax
6
Tb (z,s)] + E(z) 6w(s)
(5.17)
The general solution of Eq. (5.15) is
6T(z,x,s)
=
m(z,s) cosh(
x) + n(z,s) sinh(
X)
Boundary condition (5.16) forces n(z,s) to zero and the remaining
boundary condition (5.17) is used to determine m(z,s).
The final result
is
C 6Tb(zs) -
d E(z)
6w(s)
cosh(
6T(z,x,s)
d
sinh(f' d) + C cosh(a
Then, for x = d, the inner wall temperature
x)
(5.18)
d)
6T (z,s) = 6T(z,d,s)
becomes
C 6Tb(z,s) -
6T (z,s)
d E(z)
6w(s)
(5.19)
=
F(s) + C
where
F(s) =
Tks
tanh V'Tk
(5.20)
*To avoid overburdening the notation, the same symbols are used for
the transformed variables, with t replaced by the complex frequency s.
-
80 -
with
T
5.2
k
=
2
d/
The Flow-to-Local-Enthalpy Transfer Function
In this section the coolant energy equation, together with the wall
temperature equation derived in the previous section, will be used to
determine the time variations of the bulk temperature or enthalpy at
any position along the single-phase region.
The general solution, for
any heat flux distribution, will be obtained first and then specialized
for three particular cases of interest here.
Some approximations to the
"exact" solution will be discussed too.
5.2.1
Solution for Arbitrary Heat Flux Distribution
The following assumptions are made:
a.)
The temperature and velocity gradients in the fluid are not
taken into account and perfect mixing and "plug flow" are postulated.
These assumptions,
probably acceptable in
turbulent water flow,
become
very poor in the extreme case of laminar flow, especially for Freon which
has a low thermal conductivity.
A pure conduction calculation using
standard methods [58] has shown that 100 seconds were required for the
temperature change at the center of a Freon column filling the test
section, initially at uniform temperature, to reach approximately
percent of a step change in wall temperature.
50
Akcasu [68] proposes an
averaged transfer function to take into account the transit time spread,
and Shotkin [14] makes a "heat source correction" based on a stationary-
III
-
81
-
fluid conduction and diffusion relation to account for the temperature
profile.
b.) The flow is incompressible; length average values of other fluidproperties are used.
c.)
There is no subcooled boiling.
d.)
The kinetic and potential energy terms in the energy equation
are, as usual, neglected.
With these assumptions, the energy equation for the coolant becomes
3h(z,t)
ff
+ G(t) Dh(z,t)
+az
6h(z,t)
=
c6T b(z,t)
and
q'(z,t)
=
h c(t) P [T (z,t)
P
q'(z,t)
A
-
(5.22b)
Tb(zt)]
is the heated perimeter, and q'(z,t)
is the linear heat input
Assuming no inlet temperature perturbation
rate to the coolant.
Tb (0,t)
(5.21)
(5.22a)
with
where
_
T*(0)
=
=
(5.23)
constant
The linearized form of Eqs. (5.21), and (5.23) utilizing Eqs. (5.22), is
1
V
b(zt)
-
+
at
36 Tb(zt)
az
=
-
[(1-a) 6 w(t)
0
0
6T (z,t) -
q'(z)
w c
6
Tb(zt)
w
0
(5.24)
T*(z) - T*(z)
w
6Tb(Ot)
=
D
0
where the steady-state terms denoted by the superscript
(5.25)
satisfy the
-
82
-
reference conditions
q'(z)
dT*(z)
ba
dz
w
0
V
h* P [T*(z) - T*(z)]
b
c
w
=
q'(z)
and
c
0
is the reference coolant velocity
w /A P .
The Laplace transforms of Eqs. (5.24) and (5.25), obtained with the
initial
condition
=
6T b(z,0)
(5.26)
0
are
36Tb(z
b z s)
-
+
6Tb(z,s)
=
-A(z)
Sw(s) + B [6T (z,s) -
6
Tb (zs)]
(5.27)
and
6Tb(0,s)
=
(5.28)
0
where the coefficients are defined as follows:
q'(z)
(1
2
w
c
A(z)
B
-
a)
P ho
c
cw
When the expression of 6T
hr*F
Iftlbm
(5.29)
1ftj
-
(5.30)
as a function of
6w and
6
Tb
from
Eq. (5.19) is substituted into Eq. (5.27), a first-order linear differential equation in
6Tb
with frequency-dependent coefficients is obtained
WAII
-
83
-
q'(z)
o
M6Tb(Z's)
b
+ K(s)
6T b(z,s)
= - _
z
where
q'
0
6w(s)
L(s)
(5.31)
gqc
is some reference (average) linear heat rate,
K(s)
+
-
V
B
F(s)
F(s) +C
0
q
[(1-a) + a F(s
w0
L(s)
+ C
[]ft
(5.32)
(Bm/r ft
(5.33)
Equation (5.31) with the boundary condition (5.26) can be solved by
a Laplace transformation in
or by conventional means to yield the
z
flow-to-local-enthalpy and the flow-to-local-bulk-temperature transfer
functions
z
H(z,s)
6h(zs)
K(s)z
--
-eL(s)
K(s)z'
q'(z')
e
e
dz'
0q0
(5.34)
[Btu/lbm]
lbm/hr
6T b(z,s)
o
(5.35)
H(z,s)lbm/hr
Knowing
6
Tb(z,s)
it is interesting to calculate the modulation of
the heat flux to the fluid.
From the conduction equation of the wall and
from the convective heat transfer equation
36T(z,x,s)
6q"(z,s)
x= d
-
h* [6T (zs)
c
w
-
6T (zs)]
b
+ 6h
c
[T*(z)
w
-
T0 (z)]
b
-
-
84
Either equation yields the flow-to-heat-flux transfer function
q"zS)
6"w(s)
Q(zs)
P
c
c
Btu/hrft
lbm/hr
5.2.1.1
F(s)
F(s) + C
H(z,s) + k E(z)]
2
(5.36)
Non-dimensional Form of the Transfer Functions
It is sometimes convenient to use the transfer functions in a nondimensional form by normalizing the enthalpy variations to
preferably at the boiling boundary)
hfg (evaluated
and the other perturbations to the
corresponding average values:
H*(z,s)
E
6h(z,s)/h
h(=)/hfg
6
6w(s)/w
o
w
--
0
hf
fg
w
0
q"(z)
6q"(z,s)/q"(z)
Q*(z,s)
0
6w(s)/w
__
o
h
H(z,s)
h* P
c - H*(zg) + a]
fg
c q'(z)
Q(z,s)
o
)
F(s)
F(s)
+ C
0
These two transfer functions are plotted in Fig. 5.2* for a set of
representative conditions with a uniform heat flux distribution (5.2.2.1).
Notice that in this figure, and in general in this work, the phase delay
rather than the phase angle is used as a more representative quantity.
Figure 5.3 is intended to remove any ambiguity on this point.
The waviness of the lines of this and following plots is due to the
limited resolution of the digital plotter used to produce these figures
directly from computer output.
--
1111mimillilil1iilllll
-
85
WIN
-
6y
(perturbed quantity)
hase
\w (flow perturbation)
Re
delay = 2Tr - phase
FIG. 5.3 DEFINITION OF THE DELAYS AS USED IN THIS WORK
It
is
also more practical to plot the transfer functions versus the
a directly measurable quantity and has convenient units
period, which is
rather than the circular frequency
w = 27/T.
One immediate observation regarding
cies,
Q*(z,s) is that at high frequen-
as the storage effects become negligible, the heat flux variations
are in phase with the flow variations, while at low frequencies, where the
wall heat storage plays an important role,
6
q"is leading
Figure 5.4 shows tie parametric variations of
6w
by 900.
H*(z,jw) in the
complex plane, around some arbitrarily chosen representative point.
effects of
5.2.1.2
G, q'
The
and z are included in this plot.
0
Characteristic Constants
The equations derived above can be characterized in the time domain
by two constants, namely the wall conduction time constant
k
(dpc
d
k/d
=
d2
&
(5.37)
-
86
-
the convective time constant
=
___d
(5.38)
h
hoc
c
h
c
and their ratio
10
T
hc
=
C
already defined.
-
=
h
The conduction time constant controls the temperature gradient in the
wall, while the convection time constant affects the transient heat
The values of these constants for this parti-
transfer to the coolant.
cular experimental setup are given in Fig. 5.2.
X = V T
The wavelength
determines the space dependence of the
In Chapter 7
solutions, which also have a periodic character in space.
the regions of unstable operation will be characterized by the values of
the ratio
zb /A,
bb
The largest
will be called the order of oscillation.
zb /X,
bb
integer not exceeding
5.2.1.3
is the single-phase length.
where zb
'bb
High Frequency Approximation
At large frequencies, i.e. when the period gets much shorter than
the conduction time constant,
F(jW)
+
L(j O)
+-
2
T <<
fTk '
(1 + j)
TkW
(1
-
a)
w
0
K(jw)
+
W
V
0
T
PAc
+
B
+
j
V
0
provided that
27T
c
>> T where
(5.39)
-
87
-
For a uniform heat flux distribution, evaluation of the integral in
Eq.
(5.34) with
K
as given above yields
L
and
z
0
q'(1 - a) V
0
H*(z,jw)
h
w
o
at
w
.
-e
~
-+-
0
j
fg
sufficiently large,
Q*(z,jw) ---
0
and
a
in a similar derivation, did not
Stenning and Veziroglu [ 7 ],
consider the wall dynamics but did account for flow-dependent heat flux
by writing
6
q"/q"
0
a
=
6
w/w
0
in
the energy equation.
the high frequency approximation presented above.
This results in
Figure 5.2 shows that
this high frequency approximation is a rather poor one in the range of
interest in this work (2 to 10 second periods).
The approximation is
even worse in the case of Stenning and Veziroglu's apparatus [ 7] which
had a conduction time constant of only about 0.1 sec.
It is suspected
that this discrepancy might have been partly responsible for the lack of
agreement between their analytical predictions and test data.
5.2.1.4
Case of Negligible Heat Storage in the Wall - Low Frequency
Approximation
It can be shown by a series expansion of the hyperbolic functions
dpcw
that when the product
mations can be made:
K(jw) -
V
0
tends towards zero, the following approxi-
-
-
88
q'
L(jW)
-
w
0
F(jW)
0
-
In practice this occurs when the period of interest becomes much larger
than
27T
(dpcw << k/d) and
k
27T
h
(dpcw << h'),
c
i.e. when, for equal
corresponding temperature perturbations, the heat flux "retained" in the
wall becomes negligible with respect to the conduction and convection
The heat flux perturbation in this case
heat flux perturbations.
Notice that the low and high
+ 0.
6q"(z,jw)
naturally disappears,
frequency approximations are identical except for the (1-a) attenuation
factor at high frequencies.
Case of Highly Conductive Wall
5.2.1.5
If
the wall is
substantial heat capacity (dpc # 0),
K(jw)
-
B
+
w
V
o
i.e. when
TTk'
.
jW
[(1-a) +
jo + l/T
0
F(jo) -p 0.
6q"(z,jw)
2
a/T
-2
w
and again
T >>
jo + l/Th
qo
L(jw)
has a
an excellent heat conductor (k + c) but still
=
h
But now the heat flux perturbation becomes
jw a q"(z) Th
0
6w(j)
w
h
0
5.2.2
Solutions for Given Heat Flux Distributions
Once the heat flux distribution
q'(z)
0
is specified, the integral
-
89
-
in Eq. (5.34) can be evaluated to yield explicitly the flow-to-localenthalpy transfer function,
H(z,s).
The solutions for three cases of
particular interest in this work are given below.
5.2.2.1
When
Uniform Heat Flux Distribution
q'(z)
0
=
0
(e-K(s)z
-L(s)
H(zs)
5.2.2.2
q' = uniform along the channel,
K(s)
Exact Chopped Cosine Heat Flux Distribution
For the heat flux distribution shown in Fig. 5.5
q'(z)
=
1n )
q'
0
where
N -
is a normalized factor and
q
Y
-cos
cos
q'
is the average heat flux.
(z)coslne
0
I'
FIG. 5.5 CHOPPED COSINE HEAT FLUX DISTRIBUTION
90
-
-
Utilizing these expressions Eq. (5.34) yields
-K(s) z
H(z,s) = -
N
L(s) e
[K(s)]2 +
[
]
0
K(s)z
K(s) sin (a
e[
1
-
('-)
0
-
[K(s)
(
sin
-T-) - (f)
0
5.2.2.3
cos (
1
0
) ]
0
cos (
0
1
-)
]
0
J
Stepwise Varying Heat Flux Distribution
This case is of special importance in this work as it approximates
closely the experimental conditions.
If the heat flux distribution is
given by
q'(z)
0
q
=
for
10
z
i
< z < Zi+1 ,
= 1,N
piecewise evaluation of the integral in Eq. (5.34) yields
(s)z
H(z,s)
=
-(s
K(s)
e
K(s)z
+
(-)
[
J-1 q'
K(s)z
31
>(-'0) [ e
j=1 q I
K(s)z.
_ e
I
K(s)zJ]}
e
eI
where
J
5.2.3
Heat Flux Distribution and Wall Heat Storage Effects
is the segment in which
z
lies:
z
< z < ZJ+l
As it will become evident in subsequent chapters, the time delays in
1111111161,
-
the single-phase region,
91
-
especially the delay between the inlet flow
peak and the peak of the enthalpy at the boiling boundary are of
paramount importance for the stability of the system.
In this section,
the effect of the heat flux distribution on the flow-to-local-enthalpy
transfer function,
H(z,s)
is examined and the effect of the heat
capacity of the wall is brought out by comparing the transfer functions
with and without the wall effect.
5.2.3.1
Importance of the Wall Heat Storage
Figures 5.6 to 5.8 show the enthalpy variations along the channel,
in amplitude and phase, for three experimental points covering the
observed ratios of single-phase length to wavelength
1, and 4).
(orders 0,
zbb/X
Both the complete solution and the high frequency approxi-
mations are shown.
These three figures emphasize the fact already noted
that the wall dynamics plays an important role, at least under the
present experimental conditions, in determining the enthalpy variations
along the channel.
For all cases there are significant differences both
in amplitude and in phase between curves
2 (with wall effect), 3 (high
frequency approximation) and 4 (no wall heat storage at all).
5.2.3.2
Importance of the Unheated Portions of the Wall
As explained in Section 2.1.1 there are unheated portions, approximately
2 in. long between test lengths.
When these cold spots were
taken into account in the transfer function calculations, using the
formulas given in Section 5.2.2.3, their importance became evident.
effect is significant, even for first-order oscillations.
differences are noted at higher modes,
The
Substantial
when the wavelength approaches the
-
unheated length.
5.2.3.3
92
-
Curves 1 of Figs. 5.6 to 5.8 show these results.
Uniform Versus Cosine Power Distribution
As both a uniform and a chopped cosine power distribution (as
defined in Table 4.1) were used for the stability experiments, it is of
interest to examine their effect on the flow-to-local-enthalpy transfer
function.
In the comparative calculations, the flow rate, period, and
average temperature,
referring to an experimental point obtained with a
uniform power distribution, were used for both cases.
The power input
for the cosine case was adjusted to give the same total heat input in
the single-phase region (same position of the boiling boundary).
Figures 5.9 and 5.10 show the results obtained.
In both cases the
amplitude distribution seems to follow the heat flux distribution;
however,
for the cosine case,
the entire phase curve is shifted down-
stream because of the low heat flux at the channel inlet.
5.2.3.4
Effects of Channel Segmentation
Finally, in order to test the validity of the simulation of a
continuous power profile by a small number of stepwise variations of the
heat flux,
as was done in
this experiment,
the transfer functions
for
the exact chopped cosine and the simulated chopped cosine were compared.
The heat flux distribution of Fig. 5.5 with
Zi = 0.25 ft and
k = 9.04167 ft
represents the case that was stepwise simulated by the "cosine" distribution used throughout this work.
the dotted line in
The exact cosine solution is
Fig. 5.9 for a zero-order case.
practically superimposed, both in
amplitude and in
given as
The two curves are
phase.
When the test was repeated for the fourth order case of Point D18A-196,
a large difference in
the results was revealed.
The exact cosine distri-
moll.
-
93
-
bution gave almost no wavy structure, while large space oscillations
existed in the simulated, stepwise cosine solution.
In order to ascertain
that the difference was not due to some computational error, the calculations were repeated with the entire channel length subdivided into 14,
28, and 56 segments instead of the original 7.
The heat flux assigned
to each segment was the average value at the corresponding portion of
Figure 5.11 shows how the solutions, although
the exact cosine curve.
widely different for 7 segments, converge very rapidly to the exact
cosine result.
For 28 segments, the two solutions are almost undistin-
guishable.
The two tests described above, namely the cold spot and the
segmentation tests, showed that small discontinuities of the heat flux
distribution might have unexpectedly large effects on the enthalpy
variations along the channel,
multiple-mode oscillations.
especially at high frequencies and with
The combined effects of the cold spots,
the
discontinuous power distribution and the wall heat storage might produce
strange variations of
H(z,jw) along the channel and even "resonances"
when the test-section-length-to-wavelength
values (Chapter 7).
ratio assumes particular
The effect of the uneven heat generation in the
conductive coating (Section 2.1.1) is certainly minor compared to the
effect of the cold spots.
5.2.4
Understanding the Physical Phenomena - A Lagrangian, Non-Linear
Solution
An attempt will now be made to gain some understanding of the physical
phenomena governing the variations of enthalpy along the channel.
-
94
-
The fact that, for a uniform and steady heat input, the enthalpy
gained by a fluid particle is directly proportional to its residence
time in the heated channel suggests that a Lagrangian description
might be appropriate.
Assuming a sinusoidal variation of the flow
+ 6V sin wt
V(t) = V
the axial position,
(5.40)
z, of a particle that spent a time
heater will in general depend on
At
and on the time
At
t'
in the
at which it
entered the heated section
t'+At
z(At,t')
V(t)
=
dt = V
0
At
-
6V
A-
W
[cosw (t'+At)
-
cosw
t']
t'
(5.41)
It is obvious that if
At
is chosen to be equal to the period
particles will reach the same point
z
in
At,
T,
all
regardless of their
entrance time, and therefore pick up the same amount of enthalpy on the
way.
Therefore, the enthalpy variations will resemble a standing wave
A = V T,
with nodes equally spaced by a wavelength
in Figs. 5.7 and 5.8 (curves without wall effect).
evident in the Bode plot of
Nodes appear whenever
as clearly shown
The same effect is
H(z,jo) at a fixed location (Fig. 5.2).
T = z/n V0,
with
n
taking integer values.
Furthermore, the slope of the linear phase relationship, given in
Figs. 5.6 to 5.8 for the case of uniform and steady heat input, indicates
that the enthalpy perturbation travels along the channel at twice the
average flow velocity.
at
z = )/2,
Considering, for example, the enthalpy variations
where they are most pronounced, this simply means that the
-
95
enthalpy-wise most perturbed particle is
peak of the flow.
-
not the one
that enters at the
It is actually the one that entered a quarter of a cycle
before and has felt the effect of higher velocity (lower transit time,
lower enthalpy gain) all the way up to
z = X/2.
The particle will reach
this location in approximately half a cycle, but only a quarter of a
cycle after the peak of the flow.
Any unsteadiness or non-uniformity in the heat input to the fluid
will tend to blunt the sharp nodes.
The heat storage in the walls
produces such an effect by delaying the heat input (Figs. 5.7 and 5.8).
A similar effect is produced by the cold spots of the heated wall.
Notice
that the delay increases twice as fast in the unheated regions, where the
enthalpy perturbation travels at the flow velocity rather than at twice
the flow velocity.
significant changes,
Fig. 5.8.
Therefore,
even a short unheated length can produce
especially when the effects accumulate,
The slow attenuation of the amplitude of
regions is due to the real part of
ing, to heat losses to the wall.
wall effect,
expK(s)
6h
in
as shown in
the unheated
K(s) (Eq. 5.32) or, physically speakThe effect disappears when, without the
becomes a pure delay.
Notice also that unheated or poorly heated regions, as considered in
Figs. 5.7, 5.8 and 5.11, can also produce delays between 0 and 180*,
are otherwise unattainable in
which
the uniform heat flux case which has the
delay oscillating between 180 and 360*.
5.2.4.1
An exact Non-Linear Lagrangian Solution
An exact non-linear solution will be developed now for the case of
uniform and steady heat input to the fluid, in order to evaluate the
-
-
For a sinusoidal oscillation of the
effects of the non-linearities.
flow (Eq. 5.40),
96
Taking the enthalpy
z(At, t') is given by Eq. (5.41).
at the heater inlet as zero,
h(O,t) = 0, the particle at z will have
gained an enthalpy
At
h(z,t) = q"
q"' is
where
the heat input per unit volume of coolant (Btu/hrft )
Subtracting the steady-state enthalpy rise
h*(z,t) = q'"
6h(z,t) = -
p
,,At*
,
where
At*
z
-
(At - z-)
V
(5.42)
(5.43)
t = t' + At
with
Equations (5.41) to (5.43), solved simultaneously for various values of
the entrance time t', will provide the variations of
location
z.
at any axial
6h
The non-dimensional form of these equations will be used
below:
z*(At*,t*) = At* +
7TV
sin(2fft* -
(5.44)
ffAt*) sin (fAt*)
0
6h*(z*,t*) = At* - z*
(5.45)
where
t*
t
At*
= At ,z*
and
6h* =
h
P
q
Equation (5.44) is used to generate values of
and variable
At*,
z*
t
for a fixed value of t*
and Eq. (5.45) provides the corresponding
6h*.
Figures 5.12 and 5.13 show the results obtained in this way for
oil
-
equal to
6V/V
0.1
1.0
and
.
97
-
Parts a.)
of these figures show the
enthalpy perturbation profiles at various times.
Thermocouples, at
fixed locations measuring the average fluid temperature would have
produced the temperature traces shown in parts b.).
At
6V/V
= 0.1
the curvesare quasi-sinusoidal and coincide with the linear solution
obtained previously, while they deviate completely from sinusoids at
6V/V
0
=
1.
Notice the progression in space and time of the enthalpy perturbation
waves in the part a.) of these figures.
Due to the linear relationship
of Eq. (5.45), points of equal transit time lie on straight lines in the
(6h*,z*) plane.
These lines are almost vertical at small
6V/V
.
At
larger relative flow oscillations, they progressively rotate around their
intersections with the
6h* = 0
axis and distort the characteristic
wave pattern.
The perturbation velocity can be obtained from the (6h*,t*) plane
by cross-plotting the
The slope of such a plot is
reaches its peak value.
velocity,
which is
constant at small
of each curve versus the time at which
z*
the perturbation
twice as large as the flow velocity.
6V/V ,
straight line, even at
6h*
This slope is
and the points deviate only slightly from a
6V/V0 = 1
which denotes a constant propagation
velocity.
Another interesting observation from these plots is that the nonlinearities, by distorting the sine waves, introduce additional time
delays; the peaks and valleys of the curves are shifted in time when the
relative oscillation amplitude changes.
-
5.3
98
-
The Position of the Boiling Boundary
The time-varying position of the boiling boundary,
primarily determined by the enthalpy variations
steady-state location,
zb
however, the movements of the
of the
BB.
6
zbb(t),
h(zbb, )
is
at the
In a low pressure system,
BB are also affected by the pressure
variations in the single-phase region.
All of these effects will be
considered in the derivation that follows.
Starting with the simplest
description, the degree of sophistication will be increased gradually in
order to separate the various effects.
5.3.1
Effect of the Enthalpy Variations
shows the enthalpy,
Figure 5.14A
hf,
h,
and the saturation enthalpy,
profiles (assumed flat in this case) around the
and at some time
t
during the oscillation.
BB,
at steady state
It is evident that, neglec-
ting second order terms,
6h*
6=
bb
-bb
dho
bb
5.3.2
Effect of the Pressure Variations
The static pressure variations will tilt the saturation enthalpy
profile, while the dynamic pressure oscillations will displace it, both
causing a corresponding movement of the
effects are considered now separately.
BB.
The static and dynamic
99
-
-
Static Pressure Effects - Variable Saturation Enthalpy Along
5.3.2.1
The Channel
With reference to Fig. 5.14B,
in
this case
61h0
6zbb
d
(dho
dh*
f
bb
dz
bb
bb
(dz
-
the superscript
the subscript
bb
and
meaning again that the quantities are evaluated at the
steady-state location of the
5.3.2.2
0
BB.
Dynamic Pressure Effects
Changes in flow will produce a change in the frictional component
of the pressure drop in the single-phase region and will add an inertial
component.
At the instantaneous position of the
pbb(zbb)
p(zbb) +
=
where assuming that
6p
and
=bb
3w
fr
(< 0)
Aplf
Apolr
lfr
z*
in
pin
constant-wl.8
6w
Ap
=1.8
lfr
6w
w
0
The inertia term is given by
in
G
t
z*
in Dt
=G -
z*
n
A
at
is an equivalent inertia length for the single-phase region
(Section 5.4.1).
neglected.
6
zbb'
is the total steady-state frictional pressure drop in
= - z.
where
pfr +
6w 5ebb
aw
the single-phase region.
6p n i
=
6
BB,
In both cases, the second-order terms (6w.6z) were
-
Figure 5.14C
100 -
shows how the combined effect of the two dynamic terms
will displace the pressure profile, and consequently the saturation enthalpy
profile, by
6h0
fbb
0
0
dhf
=
dp
pbb
fr
in
causing a final movement of the boiling boundary equal to
0
+ 6h0
bb
fbb
d
dh*
dho
f b
dz bb
-6h
bb
-z
(5.46)
-dzbb
5.3.3
The Flow-to-Position-of-the-Boiling-Boundary Transfer Function
Equation (5.46) can be written in more detail as
dhf
+ (-)
-6h*
bb
q'(zb)
bb
w
where
Shbb
dh
(-)dp
=
6
dh
dp
o
h(zbb
=
dp
6w
o
--
[1.8 Ap*
pbb
p
dp
dz
bb
lfr w0
z*0
3G
in at
z'
bb
,t)
derivative of the liquid saturation enthalpy
Pbb
evaluated at
pb
(fluid property)
dp1
(-
1
Czzbb
total differential pressure drop at
zb
b
(< 0)
The variations of enthalpy at the time-varying position of the
are given by
BB
Mftki IJ",
I111IMMINIM
4 4MIN
111111h
"I''JI11,1141111"'AIN
101
-
6h
h(z
bb
h*(z*)
bb
bb
= 6h
IMI,
-
bb
+ ( dh
dz
and can amount to a substantial fraction of
6hbbb
than
6
bb
6hb
bb$
zbb
or even become larger
ree
eoelre
when the dynamic terms dominate.
The dynamic corrections are generally small or even negligible,
except when the non-boiling length becomes very large and the period is
short.
This is the case for the higher-order oscillations.
Cast as a transfer function, Eq. (5.47) becomes
Z(jW) =
-H(z* ,jW) + (
) b
bb$
dp pbb
6 zbb(jW)
P
(jW)
lw
6w(jW)
dh
q'(zbb)
w
o
dpp p
b
dp
0
bb
dz
z*
bb
ft
lbm/hr
(5.48)
where
(jW)
P
Ap*oz
lfr
= -1.8
-
in
.
J
1
fthr
0
Figure 5.15
shows
in the previous section,
"pathological" cases,
in
6h
6h
and
6zbb
in the complex plane.
As seen
is generally located, except for some
the upper half plane;
always be in the lower half plane.
the delay in
6zbb
6zbb
will therefore
The dynamic terms will always increase
if this vector is located in the fourth quadrant
(Fig. 5.15B), while if it
is
located in
the third quadrant,
the outcome
will depend on the relative magnitudes of the friction and inertia terms
(Fig. 5.15A).
effect on
6
The static pressure term has only an important attenuating
zbb*
-
5.3.3.1
102
-
Remarks
Continuous variations of the two enthalpy profiles,
and
h*
were assumed in the derivation of the transfer function
BB,
around the
h*
presented above.
As this is not the case at the limits of the heated
and unheated segments, the solutions obtained are obviously not applicable
there.
Moreover, as within
amplitude of
Z(jW)
the unheated segments
is zero, the
dh*/dz
is limited only by the remaining lesser term
BB
Therefore, if the steady-state position of the
dh*/dz.
f
happens to be in an
unheated segment, its oscillation will be greatly amplified.
5.4
The Pressure Drop in
It
is
defined in
the Single-Phase Region
important to recall here that the single-phase region, as
this chapter has a time-varying length.
variations will be due to the change in
already mentioned in
SAp,
=
1op-
5.3.2.2.
The pressure drop
length and to the dynamic effects
Summing these effects:
6ApJ + &ApJ
1we
lz +
dp*
o
= ( d)
dz b
wo
- -bbz + [1.8 Ap*
lfr
A
bb
z*
in
t
6
w0
(5w
(5.49)
It is of course implied here that the friction factor, obtained from
steady-state experiments, is applicable to a dynamic situation.
There is
recent evidence that this assumption is justifiable at low frequencies,
1000 (R is the pipe radius and
WI/V
R
when the dimensionless parameter
v
is smaller than approximately
the kinematic viscosity) [71].
-
103
-
Using transfer-function notations
6Ap (jw)
zbb + Plw 6 w
- P z
where
6Ap 1
P
is
=
lz
6
dp*
z
-)
(dz
lz
bb
((550)
0
z*
bb
the position-of-the-boiling-boundary-to-pressure-drop-in-the-single-
phase-region transfer function,
and
SAp
P
is
lw
()
=
6w
w
=
1.8
w0
A
the flow-to-pressure-drop-in-the-single-phase-region
Both of these transfer functions appear in Eq.
5.51)
(i
jjlfr
transfer function.
(5.48).
The Equivalent Inertia Length
5.4.1
It is convenient to define an equivalent inertia length for the
single-phase region by writing
z*
in
z*
-
E
11
where
z'
1
and
A.
1
are the lengths and cross-sections of the various
channel segments contributing to the inertial pressure drop.
Although there were practically no frictional losses in the preheater
(HX2), pressure recordings at station
0 (inlet plenum) during
violent flow oscillations showed perturbations of the order of
0.1 psi,
obviously due to the inertia of the fluid in the preheater and associated
piping.
Correlation of the peak pressure perturbation to the maximum
-
104
rate of increase of the flow (dw/dt)
-
was consistent and permitted the
assignment of an equivalent inertia length of
rig.
z*
in
0.8 ft to this part of the
Adding the length of the unheated region between stations
0
and
becomes
z
in
=
2.3 ft + z*
bb
(5.52)
1,
-
105
-
Chapter 6
DYNAMICS OF THE TWO-PHASE REGION THE LAGRANGIAN ENTHALPY TRAJECTORY MODEL
The purpose of this chapter is
to establish the basic hydrodynamic
equations that will permit calculation of the pressure drop in the twophase region under time-dependent flow conditions.
The method developed
here together with the single-phase dynamics derived in Chapter 5, will
be used in Chapter 8 to investigate the stability of the entire boiling
channel.
A new method of subdividing the channel by points of constant enthalpy,
and a Lagrangian description lead to a model that permits considerable
economy in computation time.
The mass and energy conservation equations
of the model are formulated in Section 6.2, the pressure drop formulas
are derived in Section 6.3, and a complete calculation procedure is outlined
in Section 6.4.
6.1
Introduction
Several models that treat time dependent two-phase flow have been
proposed in the past, and have been shown to predict with various degrees
of accuracy the transient conditions in a boiling channel [5,13,37,38].
The common characteristic of these models is that they subdivide the
boiling length into a number of space nodes and solve the three basic
equations (mass, momentum and energy conservation) for each node, thus
-
106
-
providing a basically Eulerian description of the flow.
will be taken here.
A novel approach
The channel will not be subdivided by space points,
but rather by discrete constant enthalpy points, and an essentially
Lagrangian description of the flow will be used.
This procedure follows
the example of Wallis and Heasley [10], who appear to be the only investigators to utilize a Lagrangian description.
alize the channel.
However, they did not section-
The present procedure of sectionalizing the channel
provides for considerable simplification in the calculation of the pressure
drop.
The model assumes locally incompressible flow; however, changes of
the fluid properties according to some reference steady-state pressure
profile are taken into account.
The dynamic calculations do not rely on
any particular procedure for prediction of the steadstate:
therefore
the reference profile can be provided with any available degree of
accuracy.
Furthermore, the use of a reference profile permits considera-
tion of the relative velocity between phases.
time
Incorporation of space and
dependent fluid properties (especially saturation enthalpy) and
vapor compressibility into such a model is desirable for low pressure
systems, but it represents at least another order of complexity, and was
not considered.
The following discussion will justify the particular
formulation of the model and delineate its advantages.
Stability considerations limit the time steps that can be taken in
numerical applications of hydrodynamic models, which assume incompressible
flow, to about the transit time in the segment.
Since velocities increase
rapidly with quality, it becomes evident that the length of a node can
be proportionally increased downstream.
It would be natural, then, to
-
107 -
This is a first hint in favor of a
define nodes of equal transit time.
Lagrangian description.
Moreover, as was pointed out by Meyer and Rose
[13] in their comparison of various hydrodynamic models, such a description has the distinct advantage that exact solutions are obtained for
the continuity and energy equations.
An inspection of the two-phase pressure drop correlations shows that,
for given geometry and fluid, the pressure drop in a segment is a function
of the mass flow rate, the length of the segment, the average quality,
and the pressure, this last dependence being a weak one.
When the
Lockhart-Martinelli correlation is used, for example, the longest part of
the computation involves the determination of the friction multiplier and
the void fraction.
Both of these quantities are essentially functions of
the quality alone.
In the pressure drop formula the mass flow rate and
length act as multipliers.
It follows then that, numerically, it would
be very efficient to deal with segments of constant average quality.
This
second observation leads the suggestion that it might be convenient to
subdivide the boiling length at all times by points of constant enthalpy.
The limitations and possible extensions of the method will be
discussed in Section 6.5.
6.2
Derivation of the Basic Equations
Consider a uniformly heated boiling channel of constant crosssectional dimensions.
As shown in
Fig. 6.1,
the upstream boundary or
inlet (i) of a "translating and expanding control volume" is defined
at any instant of time as the plane at which the enthalpy of the flowing
-
mixture has a given constant value
108
h..
-
An example of such a constant
enthalpy plane is the boiling boundary at high system pressures (i.e. when
the saturation temperature remains essentially constant along the channel).
Shaded area is the control
volume at time t
FIG. 6.1 DEFINITION OF COORDINATES
Denote
z.(t)
the inlet coordinate of the control volume in an
absolute frame of reference (e.g. measured from the heated length inlet)
and
C
the relative coordinate with respect to a moving frame of
reference attached to the upstream boundary of the control volume.
Then the absolute coordinate of any plane is given by
z(t) = z.(t) + C(t)
109
-
-
Following Zuber's nomenclature [72-74] a "center of volume" plane is
defined as the plane from which a moving observer sees no net volume
The velocity of this plane is then the velocity of the center
flow.
of volume or volumetric flux density,
j,
which also happens to be
the unique velocity of the homogeneous model:
j =
+ aV
(1 - a)V
In the Lagrangian description that follows such planes and the quantities associated with them
t
will be identified by two variables, the time
and the entrance time t.,thetime at which the plane was coincident with
the upstream boundary.
The Continuity Equation
6.2.1
Consider now a center-of-volume plane that passed by the upstream
boundary at time
t. (entrance time).
be at some point
G(t,t
At any instant of time
t > t.,
At time
this plane will
), having sweeped or generated a volume
AG.
t, the total rate of increase of this volume
equals the net volumetric flow entering at the inlet boundary (i) plus
the expansion due to evaporation
C(t,t
where
j
j
Gtt)=A
A
1
+
(t
i~
=
)
q' vfg
h
hfg
(6.1)
is the center-of-volume velocity in the moving frame of
reference
dz.
j
(t)
Gi1
=
j.(t)
-
(6.2)
-1
dt
t
-
110
-
It is important to notice here the fundamental underlying assumpIt was assumed that the heat supply
tions made in writing Eq. (6.1).
rate to the fluid is constant and that this heat is immediately used for
Thus, the effects of the heat storage in the walls, variable
evaporation.
heat transfer, and thermal non-equilibrium are not taken into account.
Thermal non-equilibrium, however, might be accounted for approximately
by the use of a corrected
v
ratio.
/h
Defining*
q' vf
= A g
h fgA
fg
(6.3)
Eq. (6.1) takes the form
(t,t)
-
-
QC(t,t )
=
j
Given the inlet center-of-volume velocity
inlet boundary at any instant of time,
and the position of the
j (t)
integration of this
z.(t),
as a function of time
C
equation gives
(6.4)
(t)
t
Qt
e
=
-Qt'
e
j
(t')
dt'
(6.5)
t.
The term
DC/Dt
in equation (6.4) represents the relative center-of-
j.
volume velocity,
at any location
reference,
z
To get the absolute center-of-volume velocity,
at time
dz.I/dt,
t,
the velocity of the moving frame of
must be added, resulting in
1
* The same notation is used by Zuber [72-74]
reaction frequency.
to define a characteristic
j,
-
j(t,t.)
6.2.2
=
111
-
(6.6)
GC(t,t.) + j.(t)
The Energy Equation
Besides the usual approximations of replacing energy by enthalpy
and neglecting the potential and kinetic energy of the flowing mixture,
a "quasi-steady-state" approximation is made here in order to obtain a
simple expression relating the quality seen by an observer traveling at
j,
the velocity of the center of volume,
Therefore it is
to time.
necessary to consider time intervals short enough so that the flaw does
The enthalpy variation across an infinitesimal,
not vary appreciably.
stationary control volume at a quasi-steady state is
dh =
w
_ q'
-
w
j
w
d
-
w
A
= q Adt
[ 1-x)v
f
+
V + x V
] dt
xv
g
)
(6.7)
In the bulk boiling region, neglecting locally the changes in saturation
enthalpy with pressure
dh
=
Combining Eqs.
dx
dt
h
(6.7)
(6.8)
dx
and (6.8),
2(x + vf
vfg
(6.9)
For a homogeneous flow, Wallis and Heasley [10] obtained the same
equation without making any approximations.
occupying instantaneously a volume
Av =
A At
=
Am (vf + x v
)
They considered a mass
Am,
-
112
-
and flowing along the channel; its enthalpy rise
dh
during
dt
is
given by
dh
=
dt
dt =q' A
Am
heat input during
Am
q' Am (v
) dt
+ x v
Am
which is identical to Eq. (6.7).
In the case of slip flow there are no identifiable mass lumps that can
be followed along the channel.
The trajectories of volumes bounded by two
"no net mass through" planes (planes moving with the velocity of the
"center of mass"*) could be considered, but enthalpy is exchanged through
such planes.
Similarly mass is exchanged through planes traveling at the
velocity of the "center of enthalpy"*.
As both these definitions lead to
cumbersome corrective terms that spoil the simplicity of the description,
they were abandoned in favor of the approximate but simple description
given above.
Solution of Eq. (6.9) with the initial condition
Q(t-t.)
x(t,t.)
=
x. e
1
yields
Q(t-t
vf
+
x(t.,t.) = x.
(e
-
-
1)
(6.10)
v fg
Then integrating Eq. (6.8) with the corresponding initial condition,
h(t.,t.) = h
,
vf
h(t,t.)
=
i
h. + h
1i
(x. +
fg
i
1
Q(t-t.)
)
(e
-
1)
(6.11)
vvfg
f
Finally, using the definition of the flowing specific volume or its
* These definitions are modeled on the definition of the velocity of the
center of volume; they are respectively the mass weighted (area fraction
times density) and enthalpy weighted (area fraction times density times
specific enthalpy) mean velocities.
-
inverse,
-
113
the flowing density
l/p(t,t ) = v(t,t.)
= Vf + x(t,t ) vfg
)e
=Vfg (x. + fg
i
=(1/p.)
e
(6.12)
1
vfg
where
=
v
+
V
x.
f
p.
fg
i
that
The control volume will be further specified now by requiring
it
be at all times bounded downstream by another plane of constant
enthalpy
h
(6.11) shows that given
Inspection of Eq.
.
there is a unique "residence time"
At = (t
e
- t.)
i
h.
and
h
satisfying this
The instantaneous length of the control volume is then given
condition.
by Eq. (6.5) for
t = t
The time-dependent position of the exit
.
boundary,
z e(t )
=
) +
z(t
(6.13)
(te ,t.),
is used to calculate its absolute velocity:
dz
dz.
dt
+
i
e
t
dt
e
t
(6.14)
at
tt
e
e
-
.=a
i
The last term is readily obtained by differentiation of Eq. (6.5):
Ot
t
- t.
e
I
=
e
-Qt
e
= At
+
RC(t ,t )
e j
i
(
e
) - e
j- (tt)
j iti
-
114
-
Substitution into Eq. (6.14) gives
dz
t
At
dz.
e
-1
_
t
e
the entrance time
At = t e
t.
1
-
,
) -
+ QG(t ,t ) + j .(t
j (t ) e
(6.15)
e
now being arbitrary, but the residence time,
t.
With this requirement in mind, the entrance time
is fixed.
index (t.) can be dropped from the exit quantities and replaced by the
e.
subscript
t = te, at
The exit mass flux at time
z = ze,
is immediately
available:
G (t ) = G(t ,t ) = G(t ,z ) = p
Eq. (6.12) for
(6.16)
(t )
(6.5), and (6.4), and
is given by Eqs. (6.6),
j
where
j
p
is given by
t = t.
Summary of the Solutions
6.2.3
Given:
At
the time interval,
the time-dependent position of the inlet boundary,
the inlet volumetric flux density,
flux,
the solutions for any time
v
MAt
x
e
=
x.
I
or the inlet mass
G.(t)
x.
the inlet quality,
e
j.(t),
z (t)
+
t
h.
are:
QAt
(e
Vfg
or enthalpy,
- 1)
(6.17)
Owl
-
115
vf
h
e
=h
pe
i
+ h
p.
=
i
fg
i
At
) (e
(x. +
-
-
1) = h
v fg
f +
x
e
h
fg
-Mat
e
(6.18)
(6.19)
dz.
(t)
j
G (t)
je(t)
=
j.(t)
j (t)
=
=
-
1
(6.20)
dt
(6.21)
pe
(6.22)
+ j.(t)
C (t)
t
e(t)
=
Ot
e
-&tI
j
e
(t')
(6.23)
dt'
t-At
z
(t)
=
dz.
dz
e
dt
(6.24)
z.(t) + C (t)
e
i
=
-
t
+ QG (t) + j
(t)
- j
(t-At)
e
(6.25)
Notice that Eqs. (6.21) to (6.25) specify all the input information
necessary for undertaking the solution for the next enthalpy-time step.
6.2.4
First-Order Approximation
The solutions summarized above, though complete, are somewhat cumbersome to use, as the evaluation of
C (t) requires a numerical integration.
Equation (6.23) can be further simplified without great loss of accuracy
-
-
116
j
by expanding linearly the inlet relative volumetric flux density,
around
= t
t
-
as follows:
At
j(t
(t)=
(
(6.26)
) + s(t - t )
where
dj .
dt
1
further justified by the previous assump-
This first-order expansion is
tion of small variations of the inlet flow during the time interval.
Introducing Eq. (6.26) into Eq. (6.23) and performing the integration,
e
(t)
(e
2
-
1) +
(e
s
-
1 -
(6.27)
QAt)
02
Introducing this expression into Eq.(6.25)yields
e
dt
6.2.5
At
dz.
dt
+
dt t
dz
t
t
(e
- 1)
(6.28)
t
Zero-Order Approximation
In some cases where even a zero-order approximation of
be adequate,
j
Eqs.
(t)
=
j
and (6.28)
(6.27)
(t
-
At)
=
constant during
reduce to
MAt
1
e
(t
-2
At
e
e
-l1
At
j
(t)
could
-
117
-
and
dz
dt
6.2.6
dz.
e
1
t
tt dt
A Remark on the Character of the Model
It is instructive at this point to calculate the change in mass
time
flux from inlet to exit of the control volume at
AG .(t)
=
-
G (t)
G.(t) =
[e (t)
p
t,
j (t) p.]
-
Using Eqs. (6.19), (6.22) and (6.23):
At
AG .(t) = -
j (t)F
)+
(1-e
dz.
-2Ant'
-eAt
G
e
e
(j.(t')
-
dt
P ij
td~p
1j(t
(6.29)
Approximated to the first order, i.e. using the definition of Eq. (6.26),
Eq. (6.29) reduces to
dz.
-WAt
AG .()el
1
Q
-ot
(- 1)
) +
At - e
.
(6.30)
dt It
With the zer-o-order approximation, Eq. (6.29) reduces to
dz.
AG . (t)
eidt
=
1
-AAt
(e
t
-
1)
p.
(6.31)
Equations (6.29) to (6.31) confirm the fact that the model is accounting
for the important transient mass accumulation and space distributed flow
118 -
-
As expected, all three expressions reduce to zero at
variations.
steady-state when
6.3
s = 0
j (t) = constant or
and
dz./dt = 0.
Calculation of the Pressure Drop
The steady-state pressure drop for a segment of length
flux
G*,
x*
and an average quality
=
-(x*0 + x0 )
2 i
e
A4*,
a mass
is given in Chapter 3
as
Apo Ap*+
ac
gr
fr
where the superscript
+ Apo
Apo
was added here to denote steady-state conditions.
Under time-dependent conditions for a segment of length
G
aneous average mass flux
x = x0
=
(G + G),
AC,
an instant-
the same average quality
and at the same pressure level, the pressure drop can be written as
Apr
Ap
=
fr
AG
G
2
Ap*
1.8
(o)
+
(
AG
O
-G
t
AC + A acG
(
(632
.32)
Once the time-independent values have been calculated, the computation
of the time-dependent pressure drop is reduced to the evaluation of the
simple expression above.
It is the fact that Eq. (6.17), giving the exit
quality as a function of the residence time, is independent of the mass
flux that makes the evaluation of the total pressure drop extremely simple.
First the mass fluxes at all mesh points
at all times and the segment
lengths are established, using the equations of Section 6.2.3.
Then the
total pressure drop at any point at any time is obtained using Eq. (6.32),
by summation of the corresponding pressure drop increments at this time up
to this point.
The procedure is
further clarified in
the next section.
-
6.4
119
-
Computation Procedure
The application of the theory outlined in the preceeding sections
will be described now in some detail.
The FORTRAN subroutine
DYNPR2
that performs these calculations is listed in Appendix F.
6.4.1
Reference Steady State
It is first necessary to determine the quality, enthalpy, density
and pressure drop distributions at a reference steady-state (values with
superscript *).
The time step
At
can be either chosen arbitrarily or
by subdividing the total reference transit time in the boiling length
into an arbitrary number of segments,
boundary
point numbers
BB
1
X1=
i/e
The inlet/exit subscripts
are replaced by the mesh
i/i+l.
3
2
A0
Starting at the boiling
the segments and their boundaries are numbered as shown
(BB),
in Fig. 6.2.
N.
A0
x2X
1
2
i
100
N+1
i+1
N
Xii 1
XN
N+l
X
1
Fig. 6.2. Time-Independent Reference Conditions in the Boiling Channel
The general equations (6.17 - 6.19) could be used to determine the
quality, enthalpy, and density distributions at steady state.
In these
calculations, properly defined local average values of the fluid
-
properties should be used.
Eq.
120
-
The length of the segments would be given by
(6.23)
AC
j*
=
1
E.
1
1
where
Q. At
E.
e
(6.33)
and the pressure drops would be estimated using some appropriate model.
This procedure, although theoretically correct, does not give good
results when the fluid properties
channel.
v
and
h
vary rapidly along the
The deficiency comes from the exponential extrapolation required
to obtain the exit quantities from the inlet quantities.
ancies in the spatial averaging of
errors.
v
/h
Small discrep-
produce large propagating
It was found that a much better procedure was to calculate first
the pressure profile by conventional means (Chapter 3),
then determine
the mesh points using the constant transit time requirement and calculate
the densities at these points,
p.
.
Effective
Q
values are then
defined by
-.
1
1
and the coefficients
STAPR2
ln(
At
1)
(6.34)
pi+l
E.
can be calculated using Eq.
(6.33).
Subroutine
was written to perform the steady-state calculations and is listed
in Appendix F.
-
121
-
Time-Dependent Hydrodynamic Calculations
6.4.2
The trajectories of center-of-volume planes that enter the "oiling
t2'
length at equally spaced times
considered, the time step being again
...
3,
,
tk, ...
t
The superscript
At.
equations will refer to the plane that entered at time
shows the progression in time and space of these planes.
i
points having the sum of their subscript
m
t
refer to the same time
will now*, be
in the
k
Figure 6.3
tk+l*
Notice that all
and superscript
k
equal to
and are situated on a diagonal line.
m
The regular pattern of these lines at steady state is distorted during
transients.
The following recursive relations, directly derived from the
formulas of Section 6.2.3, specify the solution of the problem; in this
first-order, machine-oriented solution, the differentials were further
approximated by finite differences.
k
k
-
ii
.k+l
Sci
.k+l
i
.k+1
k
C
where
dz)
(dt i
(6.35)
-
(6.36)
zk+)
dt i
.k
-i
At
i
=
E. jk
C
1
(6.37)
+
i
(6.38)
B. s
i
as defined by Eq. (6.33), and
E.,
Q.At
1
e
B.
1
--
G.
At
Q(6.39)
e
02
E. -At
1
lines of equal enthalpy
tk+2
k+1
ENTHALPY
FIG. 6.3
(subscript i)
and SPACE
TRAJECTORIES OF THE CENTER-OF-VOLUME
(AND CONSTANT-ENTHALPY) PLANES
hi
-
123
-
are time-independent coefficients.
=
k
jk
+
i+1
i
k
i+1
i+1
G.
1
2
Ap k
=
AC. k.
i
*k
i+1
i+1
(6.41)
G. +)
+
(Gk
L
(6.40)
i
(6.42)
i
1.81
0
A
.
Apfr
Pfri
o )
k
G.\
Gi
k
-k]
-k+1
--Gi
G.
G.i
Ap0gri
.
A
)+
-k
G.
kk
At
-
AC i
2
+ Ap*
aciG
k
dz
dt i+1
k
z.
i+1
=
=
(6.43)
k+1
dz)
dt i
k+1
z.
+
i
+
E( s
i
6.44)
1
k
AC;.(.5
(6.45)
i
The following three rules specify the time in the variables above:
I.
For quantities referring to a point (j,
j
G,
z,
dz/dt)
and
the time index is obtained by adding the superscript to the
subscript.
II.
For quantities referring to a segment (s, AG
,
G,
and
Ap),
unity must be added to the sum of the subscript and superscript.
III.
Time-independent quantities have no superscript (or superscript
Inspection of the set of equations shows that all the information
needed to undertake the next,
(i+l)th
enthalpy step is
provided by
).
-
Eqs. (6.40), (6.41), (6.44),
6.5
124
-
(6.45), and (6.46).
Comments on the Method
By now the reader will be aware of the fact that the enthalpy
trajectory model,
for feedback.
as presented in
this chapter,
Given the inlet variations,
has no built-in provision
boundary conditions can be
rigorously specified at the boiling boundary only.
Then the conditions
are calculated at any other point downstream.
The few limiting assumptions made during the derivation of the basic
equations, namely the assumption of slow flow rate variations and the nondependence of the local saturation enthalpy on dynamic pressure variations,
suggest that the model will probably be most useful for moderate flow
Such conditions prevail at the
transients around an equilibrium point.
incipience of flow oscillations.
An increase of the system pressure is
certainly beneficial to the performance of the model.
However, the model
is better than other existing models for low pressure systems, as it
takes into account at least the steady-state pressure profile.
Notice that when the position of the
BB, as determined by the
analysis of Chapter 5, is used as the first mesh point, the pressure
dependence of the saturation enthalpy is automatically taken into account
at this point, and approximately at points downstream.
Indeed,
statically, an increase of the pressure level shifts the entire pattern
of enthalpy mesh points downstream.
the correction of the position of the
A similar effect is produced by
BB
according to the static and
-
dynamic pressure variations.
125
-
Obviously, the validity of this correction
deteriorates with distance from the
BB, and reverses into an error at
the channel exit, where the pressure is externally fixed.
A correction
that was applied to the pressure drop calculation near the exit of the
channel is discussed in Chapter 8.
It is believed that with the two
refinements mentioned above, the model is also capable of approximately
considering the time variations of the saturation enthalpy.
Unfortunately, it is impossible to account for variable channel
geometry, non-uniform heat flux profile, time-dependent heat input, or,
in general any space distributed input variable, without spoiling the
inherent simplicity of the equations. On the other hand, the effect of
local flow restrictions can be superimposed easily on the uniform geometry
channel results.
The necessity to determine the conditions at a given space point indirectly, by interpolation of values at bracketing points, constitutes
another minor disadvantage of the method.
It should also be noted here
that to be able to determine the conditions at the channel exit it is
necessary to virtually extend the boiling length.
In spite of these relatively minor limitations the model seems to
be well-suited for calculations of pressure variations for small or
medium amplitude flow oscillations and transients.
-
126
-
Chapter 7
PRESENTATION AND DISCUSSION OF THE STABILITY EXPERIMENTS
One of the primary goals of the project for which this investigation was initiated was to produce a complete and reliable stability map
that could be used to test various existing or new analytical models such
as the model described in Chapters 5, 6 and 8.
In fact, most of the
stability data presented in the literature are either isolated points or
small collections of points that do not permit thorough testing of a
model.
Moreover many experiments were conducted with poorly defined
systems, having uncertain or varying boundary conditions.
Because of
the risks of physical damage to the test sections, generally the experiments were terminated near the threshold of stability without entering
deep into the unstable region.
Therefore, few limit cycle data are
available.
Crowley, Deane, and Gouse [32] obtained the stability boundaries of
the present system at four power levels with a uniform heat flux distribution.
The frequencies of the phenomena they observed confirmed that
these were density wave oscillations.
Relative to the previous work,
the purpose of the present stability experiments was to:
a)
extend and complete the uniform heat flux stability map toward
lower power levels and higher subcoolings,
b)
and
produce a corresponding map with a cosine heat flux distribution,
.iIIII
-
-
M
illld
llkil
il
ls
IlliHIM
1111il
127 -
c) enter deep into the unstable region and study the characteristics
cf the limit cycle.
Section 7.1 describes the experimental procedure.
The difficulties
encountered in the definition of the threshold of stability are discussed
in Section 7.2.
A short survey of the various ways of presenting the data
is given in Section 7.3.
used.
Section 7.4 describes the data reduction methods
Stability maps are presented and discussed in Section 7.5, and the
period of the oscillations is considered in Section 7.6.
The effect of
the cosine heat flux distribution is examined in Section 7.7.
The
characteristics of the limit cycle are described in Section 7.8.
Section
7.9 deals with a series of experiments that were designed especially to
show the existance of standing enthalpy waves in the single-phase region.
Conclusions are drawn in the last rection of this chapter.
A total of 23 routine stability runs ("D" runs) were made.
Excluding
the first three trial runs, eight runs were made with cosine and the
remaining twelve with uniform heat flux distribution.
Another eight
stability runs provided tape recordings of the flow signal that could be
analyzed for frequency content ("TR" runs).
Finally six special runs
("E" runs) demonstrated the existance of higher-mode enthalpy waves in
the subcooled region (Section 7.9).
A total of 750 points, including 175 threshold and transition points,
was recorded.
This large amount of data required machine processing.
The
introduction of data handling programs, early in the data reduction phase,
permitted the mainpulation of large amounts of data, with any desired degree of sophistication and with minimal numerical discrepancies.
The raw
stability data and relevant calculated parameters are tabulated in
Appendix E.
-
-
128
Experimental Procedure
7.1
Crowley, Deane, and Gouse [32] have shown that the oscillation
characteristics were practically independent of the number of channels
This is
operating in parallel.
a direct consequence of the constant-pres-
sure-drop boundary condition maintained by the large bypass.
Dynamic
pressure recordings at station 0 have, however, shown a small oscillation due to the inertia of the liquid in the preheater
5.4.1).
HX2 (Section
To eliminate the weak coupling introduced by this pressure drop,
all stability experiments were conducted with a single channel, namely
channel C.
V12.
Valves
The other two channels were isolated using valves
V13
V10 and
Vll and
were left wide open for all the stability runs.
For a given test fluid and geometry, the conditions in the boiling
channel are determined by the five independent variables listed in
Table 7.1.
Table 7.1
Range of the Independent Variables During the Stability Experiments
Total pressure drop,
atmospheric
p
Channel exit pressure,
approximately 7.5 to 3.5 psi
Ape
o
Gross average linear heat rate, q'
0
Heat flux distribution, q'(z)
265, 530, 795, 1060, and 1325 Btu/hrft
(100, 200, 300, 400, 500 W/test length)
uniform and cosine
0
Inlet temperature, T040
to 125 0 F
ill
11
J11,
-
129
The inlet average mass flow rate,
-
w ,
the exit quality,
as well as the limit cycle characteristics (frequency,
amplitude,
0)
x ,
T,
etc.,
and peak flow
are considered as dependent variables.
Crowley, Deane, and Gouse [32] conducted their experiments with a
fixed pressure drop across the test channels, while varying the inlet
temperature.
The experimental path obtained in this way is parallel to
the stability boundary over a large range of temperatures (Fig. 7.1 and
Ref. [32]).
In the present experiments, the inlet temperature was main-
tained constant while the pressure drop, and consequently the average flow
rate, were varied.
Thus a path perpendicular to the stability boundary
was generated, except for the lowest subcoolings.
Since previous studies,
e.g. [9], have demonstrated that the stability of the system is independent of its history or the path followed in approaching some point, the
two approaches are equivalent.
The present approach, however, has
operational advantages.
In the previous experiments, the power region from 380 W/test length
up to the maximum safe heat flux had been covered.
It became immediately
clear that at lower power levels, with forced circulation, the system was
stable.
With forced circulation the minimum available
by the hydrostatic head in the bypass.
Ap
e
is given
It was then necessary to change
the operating procedure and conduct blowdown tests as described below.
7.1.1
Blowdown Tests
The loop was prepared for data collection by a procedure similar to
that used for the steady-state experiments (Section 4.1).
The steam and
-
130
-
water flow rates to the test section preheater
HX2
were then adjusted,
to bring the mixture to the desired nominal Freon inlet temperature.
The bypass
Data collection generally started with forced circulation.
valve V9 was wide opened and the flow through the bypass was reduced to
a minimum using valves
Vi, V2 or V3, and
V6.
Then valve
Vl
was
completely closed so as to isolate the test sections, and the pump was
stopped or simply circulated Freon in the pump bypass.
The reserve of
fluid in the large bypass was sufficient to maintain the flow in a single
channel for 15 to 60 minutes, depending on the inlet temperature and the
heat input.
p0,
During this period the level in the bypass, and proportionally
decreased continuously so that the system was led through an infinite
number of quasi-steady states.
have no effect on the stability.
with increasing
velocity through
p0
The rate of decrease was slow enough to
This was verified by taking data
(slowly filling up the bypass).
HX2
Moreover, the
was so small that the Freon inlet temperature was
indistinguishable from the heating water temperature and could be controlled
very easily.
Besides providing the necessary low inlet pressures and a
way of continuously varying the flow, the blowdown procedure provided also
for accurate flow rate measurements as explained in the following section.
7.1.2
Data Collection
During the experiments the inlet temperature,
recorded on the thermocouple recorder.
T ,
was continuously
The linear flow signal, the
integral flow signal (Section 2.2.2), and selected pressure signals from
various stations along the channel were recorded on the Visicorder.
-
131
-
Occasionally the signals from void gauges (Section 2.2.5) and thermocouples were also recorded.
An additional potentiometric strip-chart
recorder (Hewlett Packard-Moseley Model 680) was used to produce a
continuous trace of the inlet pressure,
po.
The signal fed to this
recorder was provided by a 20-psig pressure transducer and amplified by
a Statham Model
readout unit.
UR5
Two calibrated bias signals could
be added to the pressure signal to keep the pen within the chart limits
while using a large amplification.
Since the cross-section of the
bypass was constant (except at the top),
the slope of the pressure trace
was exactly proportional to the mass flow rate, regardless of the tempThe glass bypass was graduated to permit visual
erature in the bypass.
observation of the level.
At steady state, the mass flow rates obtained
"barometrically", as described above, agreed within 5 percent with the
venturi readings.
With large oscillations and at very low flow rates,
only the "barometric" method was reliable.
The top of the bypass had an
irregular shape and made flow rate measurements impossible at the beginning of the blowdown.
(valve
flow.
V15
It was then necessary to use the downcomer level
to the release tank shut) to volumetrically determine the
For this purpose, the transparent downcomer was also graduated and
the liquid level was read and noted in short intervals.
This method was
less accurate because of the delay of the vapor in the condenser.
p
trace was compared many times during every run to the manometer
level and the gauge
were smaller than
P93
0.05
pressure readings (Fig. 2.2).
The
M3
The deviations
psi and therefore negligible.
The three recorders and the manual readings were synchronized with
appropriate time marks on the charts.
During most of the runs (D4 - D16)
-
132
-
the rapidly varying signals were recorded intermittently with a large time
scale to measure accurately the characteristics of the limit cycle.
After
the discovery of higher modes of oscillation, continuous recordings at a
reduced paper speed were made in order to detect all the transition points
D17 -
(Runs
7.2
D23,
TR,and E).
Determination of the Threshold of Stability
When a boiling system is brought into the unstable region by a
continuous variation of some independent parameter, the experimenter faces
the difficult task of defining the threshold of stability.
In
theory,
a
system becomes unstable when infinitesimal oscillations around an
equilibrium point are indefinitely sustained.
system
As
in
a real boiling
the small oscillations are masked by random noise, such a criterion
is generally not applicable unless the oscillations grow rapidly.
Furthermore a train of decaying oscillations triggered by external events,
near the threshold of stability can be easily mistaken for a steady
oscillation.
applied in
It becomes clear, then that some arbitrary criterion must be
defining the threshold of stability.
Often the decision is
made by simple visual examination of the traces; e.g. Berenson [22].
Massini et al. [24] also detected the threshold by visual inspection of
pressure recordings.
They note that "the theoretical threshold power is
probably overestimated,
but by no more than 1-2 percent as it
the diagrams of oscillations amplitude vs.
amplitude to zero".
grow linearly.
There is
no assurance,
power,
appears from
extrapolating the
however,
that the oscillations
Various altempts have been made to systematize the
Eli
-
procedure.
133
-
Jain [23] states that "in some cases the inception point is
readily distinguished, ...
not observed."
, but in other cases, a clearcut threshold is
The knee of the amplitude curve versus some independent
variable can be used in the former case, while statistical methods are
generally necessary in the latter.
For example, Bogaardt et al. [21]
plot the rms value of a fluctuating pressure signal, while Mathisen [40]
defines the threshold by extrapolating toward zero the inverse of the flownoise variance.
Kjaerheim and Rolstad [26] define a "noise function"
resembling an autocorrelation function and extrapolate the fast rising
portion of this curve backwards toward zero.
For the present experiments, the threshold of stability was fairly
well defined at moderate and high subcoolings as there was little flow
noise and the amplitude of the oscillations grew very rapidly.
(Fig. 7.16A).
At low subcoolings, however, it was difficult to define the threshold by
visual observation of the traces alone.
An attempt was made to define the threshold at low subcooling by
statistically analyzing the flow signal.
signal were made,
Tape recordings of the flow
the analog signal was sampled and converted to digital
data, which was analyzed numerically for frequency content by computer.
The elaborate analysis yielded no better definition of the threshold and
therefore was not used extensively.
The threshold of stability is
defined in this work as the point where the decreasing flow trace exhibits
for the first time sustained periodic oscillations with a characteristic
frequency and a distinguishable amplitude.
The subjectivity of this
definition might be responsible for some scatter in the threshold points.
-
7.3
-
134
Similarity Considerations and Non-Dimensional Parameters
A brief survey of the various non-dimensional parameters involved
in the thermal-hydraulic dynamics of the boiling channel will be made
here to prepare the ground for presentation of the stability maps in
terms of these parameters.
As a single fluid at atmospheric pressure was used throughout the
present experiment.s, the fluid-related parameters need not be considered
here.
In more general studies these parameters include the ratio of the
critical pressures, the liquid to vapor density ratios at a given
pressure, the Lockhart-Martinelli parameter
[52]
L/D,
property index
is fixed.
(p f/y )0
2
(P /Pf ).
Xtt
[49], or the Baroczy
Moreover, the geometry ratio,
For small variations of the liquid viscosity, the inlet
Reynolds number becomes a direct measure of the flow velocity.
Many authors agree on the importance of the relative position of the
boiling boundary
(BB).
It seems then natural to use the single-phase
length to total channel length ratio,
enthalpy to total enthalpy ratio,
L /L
L /L
Ah /Ah
[75] or the single-phase
[7],
which is identical to
for a uniform heat flux distribution.
The exit quality is itself a non dimensional parameter [76], but it
can be advantageously replaced by the inlet to exit density ratio,
P.~in'/p ex
or the inlet to exit velocity ratio,
The temperature of the fluid as it
V. /V
in ex
enters the heated channel can be
replaced by the inlet subcooling or, better yet,
respect to the boiling boundary,
the ratio
Ah /h
[76,77].
[7].
[]
by the subcooling with
which can also be represented ly
-
135 -
All attempts made in non-dimensionalizing the heat input (e.g.
[15,75,77]) lead to parameters that are proportional to
q/woh
where
The analysis of section 5.3.2.1 showed that
q is the total heat input.
for low pressure systems, this ratio should be augmented by
L
dh
dp1
(y)o
h fg
fg
dp
~
change in
to account for the vapor generation due to the
(-)o
p.b
bb
dz
zb
bb
saturation pressure.
In the present case this corrective term
is practically constant, and for this reason it is omitted.
All the parameters considered so far account mainly for thermal
effects.
The momentum effects are represented by the ratio of the
pressure drop in
the single-phase region to total pressure drop,
Ap 1 /Ap.
Another parameter that emerged from the analysis of Chapter 5 (Eq.
is
5.49)
the ratio of the period of the oscillation to the characteristic
single-phase inertia time constant
T
T.
in
1.8
lfr
2Tr
G* z*
in
Stenning and Veziroglu used a similar parameter in their analysis [7].
The important time delays are discussed in detail in Section 7.6.1
For the single-phase region the "order of oscillation" is defined as
z* /V T
bb o
where
V T = X
is the enthalpy oscillation wavelength used in Chapter 5.
It seems that both the hydrodynamic and the thermodynamic similarity
conditions, mentioned above, must be met in order to have two completely
similar states of the boiling channel.
As these are incompatible in
general, there will be no single curve to represent, for example, the
-
136
-
threshold of stability, but rather a family of curves (recall that three
independent variables define the system, namely
q).
p
or
w
T ,
,
and
The dimensionless parameters actually used in reducing the experi-
mental data are given in the following section.
7.4
Data Reduction
The raw experimental data, i.e. the atmospheric (condenser) pressure,
the inlet pressure readings, the average flow rate, the inlet temperature,
the power level and distribution, the period of the oscillation, and the
peak flow (the last two from the Visicorder traces), were fed to computer
programs that calculated:
the pressure at the inlet of the heated section,
the inlet
subcooling,
.
AT
= T
sun
sat (p 1 ) - T o
the average position of the boiling boundary,
the average pressure at the
z*
bb
pb
BB,
the average subcooling with respect to the
the average exit quality,
p1
BB,
AT
subb
=
T
sat
(p0 )
bb
xex
and various non-dimensional parameters of interest:
the enthalpy paramete-,
the single-phase length,
q/wdif where
h
= h
(pe )
zb /L
bb h
Lie single-phase pressure drop,
the single-phase enthalpy,
Ap1 /Ap
=
Apobb
oex
Ah1 /Ah
the single-phase transit time or "order of the oscillation", z b/V T
The heat losses were not taken into account as they were small and
approximately
constant for each power level.
Subcooled boiling was
-
T
o
-
137
-
ignored as it has been shown in Chapter 4 that it contributes very little
to the pressure drop.
The following procedure was devised to determine the position of the
BB:
Using the measured inlet pressure as the starting point, the pressure
drop and the corresponding temperature rise for each test length are
calculated.
When the mixed-mean temperature exceeds the local saturation
temperature, successive linear interpolations permit the determination of
the position of the BB.
The method is reliable as the pressure drop in
the single-phase region can be determined with sufficient accuracy.
Subroutine SUBCBB performs these calculations and is
listed in Appendix F.
The cold spots were not taken into account for the experimental data
reduction, since
7.5
a simplified version of SUBCBB was used.
Stability Maps
The experimental threshold of stability data, tabulated in Appendix E,
and some characteristics of the limit cycle will be presented graphically
by means of stability maps.
in the
q
o fg
,
Figures 7.1 to 7.5 are typical stability maps
) plane.
AT
Each curve on these figures represents a
subb
run at constant inlet temperature.
As the experiments were conducted
with decreasing flow, data collection progressed from left to right.
The
first occurrence of oscillations or the "threshold of stability" is
denoted by circles.
The points where, at rare instances, the oscilla-
tions ceased are marked by a cross.
The deviation of the points at very
low subcoolings toward the left is due to the presence of air in the
system.
When the Freon was not exhaustively degassed, gas bubbles started
-
flowing out of the preheater
138
(HX2)
-
at temperatures approximately 10'F
below the local saturation temperature and precipitated the occurrence of
instability.
7.5.1
Higher-Mode Oscillations
At high subcoolings and low power levels the system exhibited an
interesting behavior, named "higher-mode (or order) oscillations" for
reasons that will be apparent later.
were found in the literature.
No reports on similar phenomena
The following observations led to the
detection of this peculiar behavior:
a)
In the (q/w
hfg
ATsubb) plane, at high subcoolings, the
threshold boundaries should normally curve toward the right [29].
a behavior was not observed;
Such
the stability boundaries remained vertical
or even curved to the left.
b)
It has been well established [7,24] that the period of the
oscillation is approximately equal to twice the "transit time" in the
channel.
In the higher-mode region the measured periods were equal to a
small, approximately constant fraction of the expected period (Fig. 7.11).
c)
There were sudden changes in the oscillation pattern and period,
named "transitions".
Often the oscillations seemed to decay, only to
diverge again with a different frequency.
for such a transition point.
Figure 7.6A shows the flow trace
At other times the transition occurred with-
out decay of the oscillation, by a mixture of the two modes over a few
cycles (Fig. 7.6B).
d)
The hypothesis that this behavior was due to nucleation insta-
bilities alone was rejected in light of the complete explanation given later.
VINN,
-
139 -
The analysis of Chapter 5 (particularly Section 5.2.4) showed the
existence of enthalpy nodes in the boiling channel, i.e. points where the
enthalpy variations are minimum when the flow is oscillating.
For a
spatially uniform heat input to the fluid, these nodes are separated by
a wave-length
X = VT .
The enthalpy perturbations resemble a standing
wave in an organ pipe (Figs. 5.7 and 5.8).
The oscillations were assigned
an order according to the relative position of the
in the first wavelength (z'b/A < 1),
"zero-order" or "fundamental."
order of oscillation
BB.
If the
BB
is
the oscillation mode will be called
In general
=
integer part of
zbb
-b
It is clear that the enthalpy perturbations upstream of the last node
before the
BB
do not contribute anything dynamically.
Therefore this
portion of the channel should not be taken into account for the calculation of the transit time.
This explanation makes the presence of
"unexpectedly" short periods legitimate.
A series of experiments devised
to confirm this behavior is discussed in Section 7.9.
When the (q/w h
,
ATsubb) plane is divided into stable and unstable
regions, and the unstable regions are further subdivided according to the
order of the oscillations, a consistent stability map emerges.
The
transition points become threshold points for some adjacent region exhibiting a different order of instability.
as expected, to the right (Figs.
All the threshold boundaries curve,
7.1 to 7.5).
For each mode, the period
of the oscillation can be closely correlated with the subcooling as shown
in Fig. 7.11.
-
140
-
Of course, what was said above does not explain why the system
exhibits higher modes and why transitions from mode to mode occur.
This
question should be answered in the context of the stability analysis;
a few remarks, however, will be made here.
The boiling channel is dynamically dissimilar for each mode.
At
higher modes the "inactive" wavelengths must be added to the length of the
unheated inlet piping,
changing the relative position of the
BB
in the
effective heated length.
There seems to be a dynamically acceptable range of periods (approximately 2 to 10 s
in the present case).
At low subcoolings, and high
power levels, as the transit time and consequently the fundamental period
become short, the occurrence of higher modes with even shorter periods is
prohibited.
On the other hand, with very low heat inputs and inlet
velocities, at high subcoolings, the transit time becomes excessively long
and the fundamental period is divided into smaller fractions.
The
transitions from mode to mode are probably due to the combined effects of
the spatial oscillations of the enthalpy along the channel, the cold spots
of the heated wall, and they are possibly enhanced by nucleation
instabilities.
As the inlet flow is gradually decreased, the period, the
wavelength and the position of the
be considered.
BB change and their combinations must
When, for example, the
BB
reaches an enthalpy node, the
oscillation might be no longer sustained and another mode might emerge.
On the contrary, the oscillations will be amplified if the average position
of the
BB
reaches a cold spot (Section 6.3.3.1).
In one instance, at very
low inlet velocity, it has been possible to observe visually a change of
-
mode.
141 -
There seemed to be a sudden shift of the average position of the
oscillating
to a few inches upstream, suggesting that nucleation
BB
effects might be occasionally triggering the frequent transitions.
It is
evident, however, that the higher oscillation modes are not nucleation
instabilities.
In the remainder of this work the oscillations are always assigned an
order, as defined in Section 7.4.
This simple parameter does not take
into account the heat flux distribution and the wall heat storage effects.
Both of these modify the spacing of the enthalpy minima as shown in Figs.
5.8 and 5.10.
When the required corrections are kept in mind, the order
of the oscillation can be very useful in categorizing the oscillations.
Extensive graphical search and cross-verification was used to draw
the threshold lines of Figs. 7.1 to 7.5.
All the experimental observa-
tions had to be used simultaneously to provide continuity of the boundaries
from all points of view.
used extensively.
Period versus subcooling plots (Fig. 7.11) were
In many instances, when there was doubt, the original
recordings were reexamined.
In the light of what was expected, features
of the oscillations that were not noticed during the initial data reduction were found.
Obviously the information in the higher-mode regions is
limited and some extrapolation and speculation is necessary in order to
plot all the boundaries.
The information, as presented in Figs. 7.1 to
7.5, however, is consistent and all the small discrepancies can be
justified on experimental or other grounds.
Regions where extensive
mixing of modes occurs are probably indifferent to the order of the
oscillation.
-
142
-
Zero-Order Stability Boundaries - Uniform Heat Flux Distribution
7.5.2
The points at the first threshold of stability, regardless of mode,
and the zero-order transition points were plotted in Fig. 7.7, together
with the smoothed zero-order boundaries.
Five power levels are represented
on this figure, all with a uniform heat flux distribution.
At 200 W/test
length (W/TL) two boundaries are shown as no decision could be made
regarding the mode of the oscillations in the region between these two
curves.
There seemed to be a continuous mixing of zero and first order
modes, as shown in Fig. 7.6C, with the flow peaks grouped two by two.
Similar behavior also occurred at other power levels.
There is a single
higher-mode threshold point at 400 W/TL and none at 500 W/TL.
The smoothed boundaries from the experiments by Crowley, Dean, and
Gouse were also plotted on this figure and labeled "C-D-G".
As the inlet
pressure was not available for the points on these curves it had to be
calculated using the methods of Chapter 3 in
order to obtain
ATsubb
Figures 7.8 and 7.9 are the conventional stability plots in the
inlet velocity - inlet subcooling plane.
plotted on these figures.
increasing heat input is
old points in the (Ap 1 /Ap,
Only zero order points were
The enlargement of the unstable region with
clearly visible.
q/w h
) plane.
Figure 7.10 shows the threshThe boundaries at all power
levels are almost indistinguishable in this plane.
replaced by
Ah 1 /Ah,
much further apart.
a similar plot is
When
Apl/Ap
is
obtained, but the boundaries are
-
7.6
143
-
Period of the Oscillation
An analytical expression for the time delays in the channel will be
developed first in this section, in order to provide a basis for
discussion of the experimental data.
Time Delays in the Channel
7.6.1
The instabilities studied in this work are due to delayed feedbacks
between the flow rate, the void fraction, and the resulting pressure drop.
A small sinusoidal perturbation of the inlet flow will be considered now
and its effects on the pressure drop will be determined.
A uniform and
constant heat input to the fluid will be assumed for simplicity.
The analysis of Section 5.2.4 has shown that, if the flow was
t = 0, the enthalpy perturbation at the
maximum at time
BB
will reach
a maximum at time
zbb
At
1
=
bb
2V0
The ratio of this time to one half the period of the oscillation has been
already used extensively as the order of the oscillation.
In the two-
phase region the enthalpy perturbationts will create void perturbations.
These,
according to Zuber [72],
propagate with the kinematic wave velocity
which is approximately equal to the vapor velocity.
phase region, the appropriate transit time is
At
2
dz
V g(Z)
=
zbb
Therefore, in the two-
-
144
-
The maximum of the void perturbation will reach the exit after a time
At
ex
At
=
1
+ At
2
The perturbation of interest, however, is the total pressure drop
perturbation, which will depend in an integral fashion upon the local
void and flow perturbations.
The period will be approximately equal
to twice the total delay,
,
At
since the delays in the two-phase
region are short, due to the large mixture velocities.
Other authors
[7,24] have used the same assumption without, however, accounting for the
fact that the enthalpy perturbations travel at twice the flow velocity.
The following expression for
At
was derived assuming homogeneous
ex
flow, constant properties and uniform heat input to the fluid.
P
At
Notice that
At
Ah ,
I/c.
1
+t
2
=
--
2 q
Ah
1
+
q
h
,,
v
v
ln(1
fgf
vf
x
ex
)
is a linear function of the enthalpy rise in the single-
At
phase region
ATsubb = Ah
ex
=
or of the subcooling with respect to the
BB,
It increases logarithmically with the exit quality and
is proportional to the inverse of the heat input.
7.6.2
Period of the Oscillation at the Threshold of Stability
The measured period of the oscillations ranged from approximately 2
to 10 seconds, increasing continuously with subcooling.
Figure 7.11 is
a typical plot of the periods measured at the first occurrence of oscillations and at the ttansition points, versus subcooling with respect to
MA,
-
the
BB.
145
-
(The period measured after the transitions were plotted.)
the points taken at the power level of
200 W/TL,
cosine heat flux distributions, are shown.
for both uniform and
The experimental points lie on
a family of curves, grouped according to their order.
with two successive order codes had values
All
zb /A
bb
Points shown
close to an integer.
For instance, both orders two and three were assigned to points having
a value of
zb bA
bb
equal to
2.95
or
3.1
to account for experimental
uncertainties.
All the zero-order period data are plotted in Figs. 7.12 and 7.13
for uniform and cosine heat flux distributions respectively.
anticipated in the previous section are evident.
Trends
The period increases
with subcooling and is roughly inversely proportional to the average heat
input.
There is generally little scatter of the data.
Cross examination
of these figures and Figs. 7.1 to 7.5 will show that the points lying off
the smoothed threshold boundaries also lie off the period curves.
Given
the assumption that there is a unique possible frequency for each point on
the stability map, this observation is not surprising.
7.6.3
Period of the Oscillations in the Unstable Region
In the unstable region, for constant average subcooling, the period
of the oscillations varied very slowly.
map showing the lines of equal frequency.
Figure 7.14 is a typical contour
The lines present
inflection points situated approximately on a diagonal line.
moves upwards at a higher heat input (500 W/TL).
the inflections disappear.
This line
At lower power levels
This behavior was tentatively attributed to non-
linearities in the boiling region and/or to flow regime changes.
-
7.7
146
-
Cosine Versus Uniform Heat Flux Distribution
Figure 7.9 shows the zero-order stability boundaries obtained at
three different power levels with the cosine heat flux distribution.
Zero-order oscillations were not encountered at
100 W/TL.
Comparison of
these boundaries with the corresponding uniform heat flux threshold curves,
shown in dotted lines, suggests that the effect of the cosine heat distribution was stabilizing at all but the lowest subcoolings.
The comparisons
made in Chapter 4 showed that there is little effect of the heat input
The major effect on stability must
distribution on the pressure drop.
therefore come from the changes in the average position and dynamic
behavior of the BB.
The latter depends on the oscillation frequency.
In
fact, the period versus subcooling plot of Fig. 7.13 shows that the curves
obtained with the cosine distribution always lie below the corresponding
uniform heat flux curves.
The explanation is given below.
The enthalpy along the channel, for both uniform and cosine heat flux
distributions, is sketched in Fig. 7.15A.
This figure shows that for
identical inlet conditions and equal average heat input, the
BB
will
be displaced toward the center of the channel in the case of the cosine
power distribution.
Consider now the enthalpy delay curve, (2fr - arg6h(z,jw),
Fig. 5.9, redrawn in Fig. 7.15B).
For any heat flux distribution peaked
toward the center of the channel the
6h
delays are shortened.
Figure 7.15B
shows how the two effects tend to cancel in the first third of the channel,
while they add up as the
BB
moves downstream.
Since the boiling delays
are small and approximately equal for the two cases, the period of the
oscillation will follow the trends exhibited by the
6h
delay.
These
I,
-
147
" 10101MIN
-
trends are evident in Fig. 7.13 where the cosine curves depart progressively from the uniform heat flux curves as subcooling increases, moving
the
BB
downstream.
The stability of the channel is influenced accordingly,
as shown in Fig. 7.9.
7.8
Oscillation Amplitude in the Unstable Region - The Limit Cycle
Figure 7.16A shows the growth of the oscillations at the threshold of
stability at high subcooling.
Only a few cycles and a small decrease in
average flow rate were necessary to bring the system from a stable regime
to the limiting oscillation cycle.
The violence of the oscillations,
measured by the peak to average flow rate, increased with subcooling.
At low subcoolings the transition from stable to unstable flow was much
more gradual as shown in Fig. 7.16B.
Figures 7.16C to F are typical recordings.
a hot wire void gauge and a thermocouple.
They include traces from
Notice the phase differences
between the flow, the pressure along the channel, and the void fraction.
The pressures in the boiling region are minimum when the inlet flow is
maximum confirming the dynamic nature of the instability.
The pressure
oscillations are more pronounced in the center of the channel.
Notice also
the resemblance of the temperature trace to the curves of Fig. 5.13B.
7.9
Experiments to Confirm the Mechanism of Higher-Order Oscillations
The statement that, in the case of higher-mode oscillations, the
portion of the channel up to the last enthalpy node before the
not contribute to the enthalpy perturbation at the
BB,
does
BB
was made in
-
148
-
If this were correct, it should be possible to remove the
Section 7.5.1.
power from this portion of the channel and provide the corresponding enthalpy
rise by an increase of the inlet temperature without affecting the oscillations.
A number of experiments (series E) were conducted to verify this
hypothesis.
An experimental transition point with a wavelength approximately
equal to the test length was chosen, in order to be able to cut the power
for portions of the channel that were multiples of the wavelength.
Obviously, such a procedure assures similitude of the enthalpy at a
single point, as the relation
Ah
=
q/w
=
c
ATsubb
is true at a unique flow rate only.
Four experimental runs were made
(Runs E03 to E06) with the power shut off from the first 0, 1, 2, and 3
test lengths, respectively.
The oscillation characteristics obtained
(data tabulated in Appendix E)
were very much similar for each run.
Figure 7.17 shows the experimental points in the
(q*/w h
o fg
,
AT*ub
subb
The corrected (asterisked) quantities are
plane.
q*
_
-
7
7-n
q
AT*
subb
n
(q/7)__
+n(q/7)
AT
=
subb
w
T
0
and the corrected order of the oscillation becomes
zbb - n AL
+ n
where
n
is
the number of test lengths
average heat capacity of the liquid in
AL
without
power and
the non-heated length.
c
the
The
transition boundaries at the corresponding power level of 200 W/TL, from
fill
-
149
-
Fig. 7.2, are shown in dotted lines.
, z* /L ) plane is used in Fig. 7.18.
The (q*/wh
o fg
bb h
Inspection of
this plot shows that, to a first approximation, the transitions occurred
whenever the
BB
was crossing particular points of the channel.
In general the experiments have shown a remarkable similarity in
the dynamic behavior of the channel, confirming the initial hypothesis.
7.10
Summary and Conclusion
Two complete stability maps, one for uniform and one for cosine heat
flux distribution were presented.
The large number of threshold points
taken permitted the drawing of stability boundaries with sufficient
accuracy.
zing.
The effect of the cosine heat flux distribution was stabili-
Although it is generally not possible to completely describe the
stability of the channel by a non-dimensional analysis, some non-dimensional parameters were very useful in presenting the data.
The presence
of higher-mode oscillations was detected and when the
unstable regions were characterized by the order of the oscillation, consistent stability boundaries emerged.
Many observations could be
explained by the dynamic behavior of the boiling boundary.
The associa-
tion of the higher-mode oscillations with standing enthalpy waves in the
single-phase region was well established.
The oscillation pattern was
independent of the history or the path followed into the unstable region.
Some characteristics of the limit cycle were presented.
There seems to be a limited range of possible oscillation frequencies.
A unique correlation between the fundamental period of the oscillation and
the steady-state conditions in the channel was shown to exist.
-
150
-
Chapter 8
STABILITY ANALYSIS
The theoretical models for the stability analysis undertaken in this
chapter were developed in Chapters 5 and 6.
The dynamics of the single-
phase region were studied in Chapter 5 and a method for predicting
oscillatory pressure drop in the two-phase region was developed in
Chapter 6.
Both analyses were based on the assumption of an oscillatory
inlet flow.
In this chapter, a stability criterion is developed and a
block diagram for the system is presented.
Then experimental data are
used to test the model and the discrepancies observed are analyzed.
As already mentioned,
of the
BB,
pbb(t), is used here rather than the pressure at the
reference position of the
BB,
the pressure at the time-dependent position
BB,
pb(t).
The pressure perturbation at the
6pbb(t), which is not a directly measurable quantity, is related to
the single-phase and two-phase pressure drop perturbations:
6
6Ap 1
6Ap2
=
pbb
-
6P00
=
6pex -
6pbb
=
6
Pbb
'bb
The second equality in these equations implies that there are no inlet
and exit pressure perturbations.
IM,,~
ii. A1114111IMMI
-
8.1
151
011''
-
Stability Criterion
Figure 8.1 shows the block diagram of a boiling channel.
The single-
phase dynamics are represented by three transfer functions, namely
P lz,
Plw,
and
given by Eqs. (6.48),
Z,
(6.50) and (6.51) respectively.
As the treatment of the two-phase region is non-linear, the corresponding
block will be called the describing function
P2 .
The fundamental
component of the harmonic response of this block will be used in the
linear stability analysis of the system.
The pressure drop perturbations
in the single-phase and the two-phase regions are summed to yield the total
pressure drop perturbation
6Ap 1 + 6Ap 2 = 6Ap
Consider an approach to the unstable region by a variation of some
parameter, e.g. the average mass flow rate.
lating with some frequency
For the inlet flow oscil-
W, the system will cross the threshold of
stability whenever
6Ap 1 (jo) + 6Ap 2 (jw)
+
0
The stability criterion is impractical to use in this form as it relies
heavily on an accurate prediction of the perturbation amplitudes.
To
permit application of conventional control theory methods, the block
diagram of Fig. 8.1 is transformed into the closed loop diagram of Fig. 8.2.
As shown in this figure, the net inlet flow perturbation produces a movement of the
BB,
zbb,
and a two-phase pressure drop perturbation, 6Ap 2 '
-
152
-
As there is no externally applied total pressure drop perturbation (6Ap = 0),
is transformed into a single-phase pressure drop perturbation, 6Apl.
6Ap 2
After
subtracting the static contribution 6Aplz (due to the change in
position of the BB) the remaining dynamic single-phase
perturbation,
is used to generate the feedback inlet flow perturbation,
6w
the single-phase-pressure-drop-to-flow transfer function,
P
-1
lw
6Ap 1 w,
through
.
The
stability of the channel is then investigated by examining the open loop
transfer function,
6
W(s)
=
wf
w
0
in a conventional manner.
For any operational point
W(s) must be evaluated
for a range of frequencies and plotted in the complex plane.
Since most
physical aspects of the problem are understood, it will be simply stated
that the channel is unstable whenever the locus of the open loop transfer
function crosses the real axis to the left of the
8.1.1
-1
point.
Range of Frequencies
As was shown in Section 7.6, excluding the higher mode oscillations,
there is a unique correlation between the period of the oscillation and
the average conditions in the channel.
Therefore it will be sufficient to
examine the stability of the system for a limited range of frequencies.
An estimate of the period of the oscillation can be obtained from the
"total transit time",
Atex , defined in Section 7.6.1.
Figure 8.3 shows
indeed that the period of the oscillation (at the threshold of stability)
MMAINIM11M
-
Ni
-
153
is approximately equal to twice the total transit time At
.
For higher
mode oscillations this correlation is still true, provided that the integer
subtracted from
order of the oscillation is
2-
-
int
[ z bb ]+
2At 2
2At .
~2
As there is no way of predicting A priori the order of the oscillations,
the range of frequencies must be extended toward higher values whenever
the occurrence of higher modes is suspected.
To account for the wall effect,
At /T
should be replaced by
1 - arg Shobb/ 2 7r. Unfortunately, the evaluation of
6
requires
hbb
a
priori knowledge of the period of the oscillation.
8.2
Computer Program for Stability Analyses
A computer program was written in FORTRAN IV to carry out the numerical work necessary for the stability analysis.
The program, listed in
Appendix F, is described briefly here.
The required inputs are:
the inlet temperature,
the heat input,
mass flow rate, an estimate of the total pressure drop,
and the period of the oscillation.
steady-state location of the
phase region.
the
the exit pressure,
Subroutine SUBCBB determines the
BB and the pressure drop in
the single-
DZDWTF calculates the single-phase transfer functions,
namely H, Z, Plz and Plw.
Subroutine STAPR2 is then called to establish
the reference pressure and enthalpy profiles in the two-phase region,
according to the methods of Chapter 3.
the perturbations at the
BB
With this information and using
provided by DZDWTF as inputs, DYNPR2 performs
-
154
-
the dynamic calculations in the two-phase region according to the enthalpy
trajectory model.
As noticed in Chapter 6, the enthalpy trajectory model
provides time-dependent information at the enthalpy mesh points, which do
not necessarily coincide with the exit of the channel.
The procedure
devised to determine the pressure perturbations at the exit is
described
below.
8.2.1
Exit Pressure Perturbations
Knowing the pressure at two points bracketing the exit, the pressure
variations at the exit can be obtained by linear interpolation (Chapter 6).
This method fails to account for the true exit pressure as saturation
enthalpy values relative to the reference profile are used at each mesh
In
point, regardless of the magnitude of the time-dependent pressure.
order to improve this situation, the given exit pressure was used to
evaluate the fluid properties in the virtually extended channel (Section
6.7).
This assures that the correct value of the pressure is used around
the exit when the mesh points situated at steady-state outside the physical
channel are pulled back into the channel during the oscillations.
however,
Conversely,
when interior mesh points are pushed outside the channel, no
correction can be applied.
the exit of the channel in
this behavior further.
Given the importance of the pressure drop near
the present system, it
became necessary to correct
This was accomplished by recalculating the pressure
drop with the time dependent value of the saturation enthalpy in
the segment
extending from some mesh point inside the channel to the channel exit.
This method also provides an artifice for approximately taking into account
-
thermodynamic non-equilibrium.
can be
155
-
Assuming that thermodynamic non-equilibrium
simulated by a change in the saturation enthalpy, a correction of
the channel exit pressure might appropriately account for this effect.
All features mentioned above were included in
the computer program.
Recalculation of the pressure drop near the exit accounted for up to 50
percent change in the total two-phase pressure drop perturbation.
The
linear interpolation scheme gave, as expected, larger perturbations except
at high mass flow rates.
Adjustment of the exit pressure to 18 psia
accounted typically for an additional ten percent reduction in the pressure
drop perturbation.
8.2.2
Phase of the Pressure Perturbations
The phase of the pressure perturbations is determined simply by
searching for the maximum calculated value (Subroutine PRTPL3).
method evidently limits the available resolution.
This
It might have been more
accurate to fit the perturbations to a pure cosine in the least squares
sense, and then determine analytically the phase of this function.
For
the present work the former method was sufficient.
8.2.3
Options and External Corrections
The computer program, being at this stage a research tool rather
than a production code, was given sufficient flexibility to permit
thorough testing of the theory.
For example,
various parameters non-
essential to the stability analysis are calculated and printed.
"Switches" allow bypassing certain calculations or repeating others with
-
partially different inputs.
156 -
It is also possible to externally apply
corrections to the amplitude and phase of the
8.2.4
BB perturbation.
Performance of the Code
Despite of the serious burdening of the program with peripheral
calculations and some lack of efficiency due to duplication of effort in
various subroutines, the program was rather economical to run.
A
complete calculation of the open loop transfer function, for one frequency
value (120 time steps, 16 enthalpy mesh points, limited printout)
required approximately
8.3
1.5 seconds on the
IBM-7360/65 (FORTRAN Compiler G).
Prediction of the Threshold of Stability
The computer program described in the preceding section was used to
predict the behavior of the channel for a number of experimencal points.
Figures 8.4 and 8.5 show the calculated open loop transfer functions.
The
frequency response obtained is similar to results reported in the literature [7,12].
The small loops
in the figures occur when the wavelength
approaches the single-phase length (zb /VTrl).
been reported by Stenning and Veziroglu [7].
Similar results have
The gain of the open loop
transfer function increases, as expected, with decreasing flow rate, as
the unstable region is approached.
The threshold of stability, however,
as predicted by the frequency analysis was not in agreement with the
experimental observations (see data included in Figs. 8.4 and 8.5).
The
curves cross the real axis for values of the period well below the measured
ones.
The agreement is, however, slightly better at low subcoolings.
11
WIIHMINMINN16
bilmillilill
-
157 -
Discussion of the Discrepancies
8.3.1
The origin of the disagreement between experimental observations and
typical theoretical predictions is discussed with reference to Fig. 8.6.
The two components of the dynamic pressure drop perturbation in the singlephase region,
6 Aplw,
component,
6 Aplfr
component,
6Aplin,
are shown separately in this figure; the frictional
is 1800
out of phase* with
while the inertial
6w,
leads the frictional component by
90*.
Im
dir. 6zbb
6AP1
Re
6Apm
.w
Re
2z
Plfr
SINGLE-PHASE REGION
T1O-PHASE REGION
FIG. 8.6 VECTOR DIAGRAM OF PERTURBATIONS AS TYPICALLY PREDICTED
BY THE MODEL
are defined here as negative quantities. In
and Ap
Recall that Ap
the opposite convention was adopted and
however,
program,
the computer
as positive quantities.
treated
were
drops
the pressure
-
158 -
In the two-phase region the pressure drop perturbation was approximately
1800
out of phase with the flow perturbation, suggesting that
the delays and the inertia effects were small.
components of
component
6Ap 2 ,
6Ap 2 w,
the static component
This plot shows also two
6Ap 2 z
and the dynamic
although the enthalpy trajectory model does not make
such a distinction.
Im
orrection to
Zbb
correction to
|6AP2wi
Re
6Ap2
FIG. 8.7
CORRECTIONS TO PREDICTED PERTURBATIONS
Figure 8.7 shows the adjustments that would have been necessary to
correct the results.
These are analyzed below and their relation to
physical phenomena is brought out.
It is obvious, that in order to have
-
6Ap 1 + 6Ap 2 = 0
159
-
it is necessary to rotate
Assuming that
counterclockwise.
6Ap1
6Ap1
clockwise and/or
6A p2
is correctly estimated, the first
correction can be accomplished by an increase in either the delay or the
amplitude of
able.
6hbb.
It is likely that the phase of
Shbb
is question-
Indeed the analysis of Chapter 5 assumed a perfect mixing of the
flow and flat radial temperature and velocity profiles.
these assumptions for Freon is doubtful.
The validity of
Unfortunately, there is little
information on transient convective heat transfer to undertake a rigorous
examination of this complicated problem.
Qualitatively, it seems that
the enthalpy variations travel with a lesser velocity than expected.
The
heat generated in the wall has to be conducted through an almost stationary laminar sublayer to diffuse to the central core of the duct, from
where it will be transported downstream.
It can be deduced that the
combined effects of the temperature and velocity profiles introduce
significant additional delay.
A reduction of the amplitude of the dynamic two-phase pressure drop
perturbation,
6Ap 2 w,
will result in a counterclockwise rotation of
6Ap 2 '
This dynamic perturbation has been generally overestimated by the enthalpy
trajectory model and is responsible for the approximately 180*
angle of
6Ap 2 .
For the range of qualities under consideration, the
frictional pressure drop is
In addition,
the order of
phase
dominant in
the transit time in
the two-phase region (Fig.
the boiling region is
3.5).
also short (of
0.5 s) compared to the order of the oscillation.
-
160 -
Therefore, little phase correction can be achieved by changes in the
space - time distribution of the flow perturbations.*
Comparison of the dynamically obtained pressure drop perturbations
with corresponding steady-state differences in the two-phase pressure
drop showed that they were of the same order of magnitude.
The exit
pressure drop correction discussed earlier in Section 8.2.1 and eventual
adjustments of the exit pressure to account for thermodynamic nonequilibrium provided insufficient reductions
of the amplitude of
Therefore
it can be
6Ap2'
argued that time-dependent heat transfer plays an
important role in determining the amplitude of the two-phase pressure drop
perturbation.
The dependence of the heat transfer on the flow velocity
will reduce the changes in the vapor generation rate and consequently
diminish the pressure drop perturbation.
came to the same conclusion.
cient
a
Stenning and Veziroglu [7]
They had estimated the value of the coeffi-
(Eq. 5.9) to be close to
0.4;
a value of approximately
0.7
would have been required for their observations to agree with their
theoretical predictions of the threshold of stability.
The importance of the effects mentioned above diminishes as the
boiling region is extended upstream.
This explains the closer agreement
obtained at low subcoolings.
It can be concluded that two simultaneous corrections were necessary
*
In the progress of the experimental work a short movie was made to
study the space - time distribution of the flow perturbations. A
system of mirrors was used to photograph simultaneously, on the same
frame, the inlet and the exit of the channel.
00101
-
1
161 -
to bring agreement between the theory and the experiments:
a) a signi-
ficant reduction of the amplitude of the dynamic two-phase pressure drop
perturbation, and
Both were explained on physical grounds but could not be incor-
region.
porated in
8.3.2
b) an increase of the delays in the single phase
the model.
Higher-Mode Oscillations and Transitions
Examination of Figs. 8.4
and 8.5 suggests that the occurrence of
transitions can be related to the presence of small loops in the locus
of
W(s).
When the real axis intercepts the locus of the open loop
transfer function at more than one point multiple modes of oscillation
are possible.
Moreover, the results of the frequency analysis shown in
these figures are in agreement with the experimental fact that at high
mass flow rates unstable operation is possible only at low periods.
Unfortunately, the discrepancies discussed above did not encourage further
analytical investigation of this problem.
-
162
-
Chapter 9
GENERAL CONCLUSIONS AND RECOMMENDATIONS
Partial conclusions drawn in previous chapters will be combined
and summarized here and recommendations for future work will be made.
9.1
Steady-State Work
Methods for predicting the steady-state behavior of the boiling
channel were presented and their validity was verified by an extensive
series of experiments.
An accurate prediction of the pressure drop in
the channel was achieved using well established correlations.
The
calculated pressure drops were particulatly accurate in the dynamically
interesting region.
The only discrepancies were due to deviations from
thermodynamic equilibrium between the phases.
brium is
Thermodynamic non-equili-
at present under investigation in various laboratories.
It
is
hoped that these studies will provide a theoretical basis for accounting
for these phenomena.
Freon-113 seems to be particularly prone to thermodynamic nonequilibrium.
conductivity.
It
has also a high air solubility and a very low thermal
These undesirable properties must be kept in mind when
future work with Freon is undertaken.
-
163
-
Stability Work
9.2
9.2.1
Experimental Results
The regions of unstable operation of the channel were mapped with
sufficient accuracy to permit evaluation of parametric trends.
A number of dimensionless groups proved to be useful in correlating
the data.
These groups often contained integral parameters that were not
directly measurable.
Evaluation of these quantities required some
numerical effort which was, however, justified by the improvements obtained
in the presentation of the data.
It is concluded that non-dimensional
groups can be profitably used for interpolation or extrapolation of
theoretical predictions and experimental measurements.
A close correlation between the period of the oscillation at the
threshold of stability and the average conditions in the channel was
established.
The period of the oscillation is approximately equal to
twice the total transit time in
the channel.
This quantity is
rigorously
defined below.
Two distinct propagation phenomena must be considered: the propagation
of the enthalpy perturbations in
the single-phase region and the propaga-
tion of the void perturbations in the two-phase region.
The enthalpy
perturbations travel at twice the flow velocity, while the void perturbations propagate approximately at the velocity of the vapor phase.
Therefore, the total transit time is defined as the sum of one-half the
physical transit time in the single-phase region plus the vapor transit
time in the two-phase region.
-
164 -
At high subcoolings and low heat inputs the channel oscillated at
frequencies that were a multiple of the expected frequency.
Sudden
changes in the period of the oscillation were also recorded in this region.
The occurrence of these "higher-order" oscillations and transitions from
mode to mode were shown to be associated with the existence of more than
one node in the standing enthalpy waves of the single-phase region.
The
"order" of the oscillation was defined as the integer part of twice the
ratio of the single-phase transit time to the period of the oscillation,
and is equal to the number of enthalpy nodes minus one.
When the regions
of unstable operation were characterized by the order of the oscillation,
a coherent stability map was obtained.
The effect of a chopped cosine heat flux distribution was investigated
experimentally and was found to be stabilizing.
Because the influence of
the heat flux distribution on the pressure drop was small, the stabilizing
effect of the cosine distribution was attributed to the dynamics of the
boiling boundary.
With the cosine heat flux distribution the period of
the oscillations was shorter than with the uniform heat flux distribution.
The difference could be explained by considering the delays in the singlephase region.
9.2.2
Theoretical Results
It was shown that heat storage in the channel walls can acquire under
appropriate conditions a major importance in the single-phase dynamics.
Under the present experimental conditions an exact treatment of the wall
dynamics was necessary to account correctly for time-dependent heat flux.
The distribution of the enthalpy perturbations along the channel is also
IJ14"11111111110wll
-
11111
165 -
influenced by local discontinuities in the heat flux distribution, such
as stepwise heat flux changes and cold spots in the wall.
The effect is
significant when the characteristic length of the discontinuities
approaches the wavelength of the oscillation,
X = V T.
The movements of the boiling boundary were determined taking into
account both the static and the dynamic pressure drop perturbations in
the single-phase region.
The relative importance of the generally
neglected dynamic terms increases in proportion to the length of the
single-phase region and also increases with the frequency of the oscillation.
A new method of calculating two-phase pressure drop under oscillatory
flow conditions was developed and shown to have several advantages over
conventional techniques in the calculation of the time-dependent pressure
drop.
The method, however, cannot account for time-dependent heat transfer
from the wall.
In the present case this seemed to be a limitation.
The single-phase and the two-phase dynamics models were used to
predict the threshold of stability of the present experimental channel.
Although the stability model exhibited a qualitatively correct behavior,
the theoretically predicted thresholds of stability were not in agreement
with the experimental observations.
The causes of the disagreement were
attributed mainly to three phenomena:
a)
radial heat conduction and diffusion effects in the single-phase
region that influence the propagation of the enthalpy perturbations;
b)
lack of thermodynamic equilibrium; and
c)
time-dependent heat transfer rates in the boiling region.
-
166
-
The prediction of excessively large two-phase pressure drop perturbations
was probably due to effects b) and c).
Recommendations
9.3
9.3.1
Experimental Work
It would be interesting to investigate experimentally the progression
of the enthalpy perturbations in the single-phase region.
Detailed
information on the radial temperature and velocity profiles under oscillating flow will be needed to test analytical models.
Fluorocarbons will
be suitable as test fluids, their low thermal conductivity enhancing the
radial phenomena.
Externally controlled oscillation of the flow will be
necessary to eliminate dependence on the natural oscillations of the
channel.
Discontinuities in the heat flux distribution should be consi-
dered with caution in future experiments.
9.3.2
Analytical Work
It is recommended that the limits of applicability of the enthalpy
trajectory model be determined by comparison to other well established
hydrodynamics codes or to experimental data.
The entire stability model
developed here should be tested under more favorable conditions, e.g.
using stability data obtained with water systems.
Two-dimensional treatment of the single-phase region is an extremely
challenging problem and should be undertaken in order to evaluate the
importance of the radial effects.
11
-
167 -
Analytical and experimental investigations of thermodynamic nonequilibrium are necessary.
-
168
-
REFERENCES
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22.
Berenson, P.J., "An Experimental Investigation of Flow Stability
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23.
Jain, K.C., et al., "Self Sustained Hydrodynamic Oscillations in a
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Masini, G., G. Possa and F.A. Tacconi, "Flow Instability Thresholds
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Bowring, R.W., and C.L. Spigt, "Seven-Rod Bundle Natural-Circulation
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26.
Kjaerheim, G. and E. Rolstad, "In-Pile Hydraulic Instability
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27.
Gouse, S.W., Jr., and C.C. Hwang, "Visual Study of Two-Phase OneComponent Flow in a Vertical Tube with Heat Transfer," M.I.T., EPL
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28.
Dickson, A.J. and S.W. Gouse, Jr., "Heat Transfer and Fluid Flow
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29.
Gouse, S.W., Jr. and C.D. Andrysiak, "Flow Oscillations in a Closed
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32.
Crowley, J.D., C. Deane and S.W. Gouse, "Two-Phase Flow Oscillations in Vertical, Parallel, Heated Channels," Proceedings of
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33.
Deane, C.W., IV, "An Experimental Investigation of Flow Oscillations
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34.
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35.
Meyer, J.E., "Hydrodynamic Models for the Treatment of Reactor
Thermal Transients," Nucl. Sci. Eng., 10, 269 (1961).
36.
Meyer, J.E. and E.A. Reinhard, "Numerical Techniques for Boiling
Flow Stability Analyses," J. of Heat Transfer, Trans. ASME, Series C,
87, 311-312 (May 1965).
37.
Grumbach, R., "A Systematic Comparison of Different Hydrodynamics
Models," Nucl. Sci. Eng., 36, 429 (1969).
38.
Bjdrlo, T. et al., "Comparative Studies of Mathematical Hydrodynamic
Models Applied to Selected Boiling Channel Experiments," Proceedings
of the Symposium on Two-Phase Flow Dynamics, Euratom - The Technological University of Eindhoven (Sept. 1967).
39.
Shotkin, L.M., "Effect of Slip Ratio on the Crucial Boiling Length
in Two-Phase Instability," Proceedings of the Symposium on TwoPhase Flow Dynamics, Euratom - The Technological University of
Eindhoven (Sept. 1967).
40.
Mathisen, R.P., "Out of Pile Instability in the Loop Skilvan,"
Proceedings of the Symposium on Two-Phase Flow Dynamics, Euratom The Technological University of Eindhoven (Sept. 1967).
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Benning, A.F. and R.C. McHarness, "Thermodynamic Properties of
'Freon'-113. Trichlorotrifluoroethane, CCl 2F-CClF2, with Addition
of Other Physical Properties," E.I. du Pont de Nemours and Co.,
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42.
Downing, R.C., "Transport Properties of 'Freon' Fluorocarbons,"
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Solid Materials," The MacMillan Co., New York (1961).
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Bulletin B-83.
46.
Hsu, Y.Y., FF. Simon and R.W. Graham, "Application of Hot Wire
Anemometry for Two-Phase Flow Measurements Such as Void Fraction
and Slip Velocity," ASME Multi-Phase Flow Symposium, 26-34
(Nov. 1963).
47.
Sieder, E.N. and G.E. Tate, "Heat Transfer and Pressure Drop of
Liquids in Tubes," Ind. Eng. Chem, 28, 1429 (1936).
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83, 503 (1961).
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MMIM
MUOMMMMNM
lik
Appendix A
PRESDROP, a Computer Code to Predict the Steady-State Operating
Conditions in the Boiling Channel
This Appendix contains a schematic description of the program
PRESDROP that calculates the steady-state operating conditions in the
boiling channel according to the procedure presented in Chapter 3.
A FORTRAN listing of PRESDROP is given at the end of this Appendix.
This program was also used in subroutine form (PRESDR) for incorporation
in any program to provide the steady-state information.
A.l
Structure of the Main Program
The program starts by reading in the general information, i.e. the
inlet temperature (TIN), the condenser pressure (PCOND), the room
temperature (TROOM), the heat input (QW(I), I=1,7) and the various
convergence criteria and control parameters.
These data are used to
perform a few basic calculations.
Next, an inlet velocity (VIN) is read in together with a guess of
the total pressure drop (PINEST).
Calculations start based on the
gross power input, and subroutine SUBBOL is called to estimate the point
of NVG.
The exit conditions are determined and the pressure drop in the
unheated exit length is evaluated.
Then a march upstream, segment by
segment, is started: subroutine SUBLIQ for negative or MARNEL for
positive exit equilibrium quality is called to perform the calculations
A-2
for a single segment.
boiling.
is
Both subroutines can take into account subcooled
When the bulk boiling boundary is reached subroutine BOBDRY
called to determine accurately this point.
the BOBDRY call is
With subcooled boiling
omitted and the zero quality point is
by linear interpolation.
At this point,
determined
the steady-state conditions
along the channel are known in a first approximation.
The heat losses
are estimated by LOSSES and the location of the NVG point is
found by
linear interpolation on the basis of the bubble departure enthalpy
provided by SUBBOL.
The calculation is
then repeated with the
corrected heat input and NVG point as many times as desired.
Finally,
printout occurs at selected points by skipping intermediate nodes.
The
program then returns to read in another inlet velocity value (VIN).
The FUNCTION subprograms called in PRESDROP to evaluate the physical
properties of Freon-113 are described in Appendix B.
Appendix C deals
with the auxiliary functions used in the two-phase pressure drop
calculations.
More information about the program can be found in the
comment cards of the FORTRAN listing.
A few convergence problems
encountered during the debugging of the code are discussed in the
following section.
A.2
Convergence Problems Around the Bulk Boiling Boundary
Subroutine BOBDRY, which is used to locate exactly the zero
equilibrium quality point, occasionally presented serious convergence
problems.
At high mass fluxes and heat fluxes, there is a very large
and sudden acceleration of the flow at the bulk boiling point, resulting
in an almost stepwise pressure drop.
This physical discontinuity,
A-3
coupled with the mechanics of the pressure drop and quality evaluations
frequently produced numerical oscillations of the pressure and quality
during the iterative process, when a narrow enough tolerance was
specified for the zero quality point.
The quality then jumped continu-
ously from negative to positive values and vice-versa without ever
approaching zero.
To circumvent this annoying difficulty, the convergence
algorithm was modified, a length rather than a quality tolerance was
imposed, and the calculations were ended when a slightly negative "zero
point" quality was obtained.
The last consideration assured that the
important acceleration term in the pressure drop was correct.
A similar problem appeared in subroutine MARNEL which calculates the
pressure drop around the zero quality point when subcooled boiling is
taken into account and the BOBDRY call is bypassed.
The problem was
dealt with in a similar fashion and to avoid expensive looping, a limit
on the number of iterations was imposed.
See the FORTRAN listings and
comments for details.
A.3
Effect of the Segment Size and Error Criteria on the Code Accuracy
Test runs with variable segment sizes (the 15.5 inch test length was
subdivided into as many as 24 segments) have shown that the pressure
profile is rather insensitive to the segment size.
Node lengths of a
few inchesused with a pressure drop convergence criterion (ERROR) of
0.001 or 0.002 give quite satisfactory results.
Even segment sizes above
ten inches give errors of a few hundredths of a psi only, mainly
concentrated around the zero quality point.
A value of about 0.02 is
A-4
recommended for the boiling boundary position error (EBB).
One major
iteration is generally sufficient to accurately estimate the losses and
the NVG point.
7
C
0R O P
C
*********
cl
11/3/69
PRESDROP CALCULATES PRESSURE DROP IN TWO-PHASE FlOW WITH VARIAML=
PRESSURE DEPENDENT PROPERTIES.
C
C
C
C
C
C
C
C
C
C
C
C
r
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
P R
INPUT
S
INFORMATION
**********
**
VERSION
*****************
q
9
10
*
*
*
*
*
*
*
*
*
*
TIN,
INLET TEMP. (F), PCOND, CONDENSER PRESS.(IN HG),
TROOM, ROOM TEMPERATURE (F),
TITLE
TO GET PRINTOUT FOR ALL ITERATIONS PUT IP .GT. 0
/3F10.5, 12A4, I?
QW(I),
POWER DISTRIBUTION BOTTOM TO TOP (WATT) /7F0.5
N, NUMBER OF SEGMENTS FOR EACH TFST LENGTH, MUST RE MULT. OF NP
NP, NUMBER OF SEGMENTS BETWEEN PRINTOUTS,
ISURC, SWITCH FOR SURBOL (POS.IF WANTED),
LOSS, SWITCH FOR HEAT LOSS CALCULATION (POS. IF WANTED),
KITER, NUMBEROF MAJOR ITERATIONS,
ERROR, CONVERGENCECRITERION FOR MARNEL (RFLATIVF ERROR)
EBB, ERRCR CRITERION FOR
BODRY (USE .GT. 0.005) /5I2,2FI0.5
11
IP,
*
* VIN, INLET VFLOCITY (FT/SEC),PINEST, AN ESTIMATE OF
*
PRESSURE (PSIA)
/2F10.5
*
* VIN,
PINEST
ETC
**** 0.0
END OF VELOCITY VALUES
/F10.5
13
14
TIN,
PCOND,
16
ETC
*
*
****0.0
0.0
EED OF DATA
C
OUTPUT
/F10.5
INFORMATION
*********
*******
*******
**
PRESSURE, TEMPERATURE, VOID FRACTION, QUALITY PROFILFS ALONG THE CHANNFL
POINT OF NVG, ROILING BOUNDARY, HEAT LOSSES, FTC
C
MAIN
PROGRAM*******
19
19
*
C
C
C
*****************VERSION-2
20
99
C
REAL MUL, MUG, MULE, MUGE, MULTF
INTEGER FIN
DIMFNSION QW(7), 0(7), HEATFL(7), P(175),
X(175), H(175), T(175),
FRI168), GR(168), AC(16R), TOT(168),
SUMFR(175),
SJMGR(1751
2
, SUMAC(175), PNET(175), A(1I5), SUMTOT(17S), XTR(175),
3
TMST(I), XST(R), QNETW(7), INET(7), TTR(175), QLOSW(7),
4
TITLE(121
COMMON D, DL, ERROR, G, OH, OTS, HE, HI, XE, XI,
PE, Pl,
BE, RI, TRE, TRI, ALFA, DPFR, DPGR, OPAC, OPTOT, XITR, XD,
2
TBITR
COMMON /SUBSUB/ QNET
COMMON /FLUID/
ROL, ROG, MIUL, MUG
COMMON /NNN/ N
1
C
1
2
3
4
5
6
FORMAT ('0',
'EXIT',3X,'0.0',4X,2F9.N,F10.,IX,2F.2,
F9.3,3X,*0.0' )
FORMAT 1'O','EXIT LIQUID PHASE REYNOLDS NUMBFR SMALLER THAN 2000,
1 RF = ', F6.1)
FORMAT(79X,4FR.3 /3X,'8',4X,'3.125',2X,2F3.5,FO.5,1X,2F8.2,
1
IX,2F8.3,2X,4F9.3
FORMAT(79X, 4F9.3 / 14, F9.3,2X,2F9.5,F1O.5,1X,2F8.2,1X,2F8.3,
1
2X, 4F9.3 )
FORMAT ('I','INLET LIQUID VELOCITY -', F7.3, ' FT/SEC', 5X,
1
'MASS FLUX, G = ', F10.1, ' LAM/HR FT2',
NX,
2
'MASS FLOW RATE, W = ', F7.1, I LBM/HR' f/
FORMAT(/f// ' BOIL ING BOIUNOARY AT Z =',
Fi.3, *
INCHFS FR
lAM EXIT, OR AT' / 25X, F8.3, '
INCHES FROM INLET'/
2
' AT ROILING BOUNDARY
EQUIL.TEMP. ='F,
7.2, '
DEG. F,
F7.2,'
DEG. E'/
,'TRUE TEMP. =*,
3
PSIA' /
F7.3, '
4
23X, 'PRFSSURE =',
5
23X, 'INLET SUBCOOLING WITH RESPFCT TO TEMP. AT ROILING
F' )
6
'BOUNDARY, TBR - TIN =', F7.2, *
******
FRROR **
******
FORMAT(///' ***
1
* INLET TEMPERATURE ABOVE SATURATION POINT' /
' INLET TEMPERATURE SHOULD RF AT LEAST BELOW ',F6.1,
2
3
'DEG F')
FORMAT(//'0FOR A PRESSURE OF',
F8.2, '
1
PSIA THE BURBLES DEPART AT TSAT-TBULK 1,
2
F6.1, '
DEG. F
'
, 'OR AT A QUALITY =', F10.5 ///)
FORMAT(////
DEPARTURE AT Z =',
F9.3, *
INCHES FR
lOM EXIT, OR AT' / 25X, F8.3, '
INCHES FROM INLET, IN TEST SECTION
2 ',
13, '
(COUNTING FROM INLET)' /
3
0 AT THE NET VAPOR GENERATION POINT THE EQUIL. TEMP. WAS',
4
F7.2, a DEG. F' /
5
35X, 'THF TRUE BULK TEMPERATIIRC WAS', F7.2, '
DEG. F'
6
35X, 'AND THE PRESSURE WAS', F7.2, '
DSIA' I
FORMAT('1',
20X, 'S U M M A R Y
0 F
R F S U I T S' //)
FORMAT('0','
VIN (FT/SEC)", '
W (LBM/HR)',
'
QWAV (WATT)*,
XEXTR
PEX (PSIA)',
'
XEX
',
1
'
2
'
ALFAEX
',
'
Pt (PSIA)',
' Pl-RHO*G*l
0,
'
OPLEX (PSI)'
3
FORMAT( F13.3, 2F13.1, F13.3, 3F13.5, 3F13.3)
FORMAT ('0',
'FXIT QUALITY LARGER THAN 1.0 , XFX = ',
F10.91
1
J
OEG.
15
THE INLET
17
****
)
4
VARIABLES AND FORMAT
****
FORMAT ('D', 17X, 'PRE$SURE DROP, QUALITY, VJID FRACTION ANO RULK
LTEMPERATURE ALONG THE TEST SECTION' /f/f
POINT', 1X, 'Z (IN)',
2
X,'K-EQ',5X,'-TR' ,X, 'VIO
,3X,'T-EQ' ,4X,'T-TR',3X,
3
'P (PSIA) P-RO*G*7
FRICTION GRAVITTY',3X,'ACCEL.',4X,'TOTAL'
FE.
*
2.1
RUBBLE
C
PRESSURE DROP FUNCTIONS **************************************
C
C
EF(X)
= (-1.6628E-11)*G**((1.0-X)**2/( I.O-ALFA)*ROL)+X**2/(ALFA
*ROG))
RELPFXI) = G*(1.0-X)*D/MUL
1
1
C
C
C
FORMAT(3F0.5, 12A4, 12)
FORMAT (7F10.5)
FORMAT (512, 2FI0.5)
FORMAT ('1', 20X, 'PRESSURE DROP CALCULATION
***
',
12A4///
* INLET TEMPERATURE = 1, F7.2, I DEG F',
1
20X, 'ROOM TEMPERATURF =', F7.1
//
2
' CONDENSER PRESSURE = ',
F7.3, '
IN HG' I/
3
' EACH TEST SECTION DIVIDED INTO ', Is, '
SEGMENTS'
If
4
' DATA PRINTED EVERY ',
15, '
POINTS'
If
5
' ERROR CRITERION = ', F10.M //
6
' EPROR CRITERION IN BORDRY =
, F10.8
/f)
FORMAT I'D', 'POWFR
If ' TEST SECTION NUMBER',
1
9X, '',9X,'2' ,9X,'3',9X,'4',9X,'5',9X,'6',3X,'7'
/I
2
POWER (WATT)',10X, 7F10.1 //
P
3
HEAT FLUX (BTU/HR FT2)', 7F10.1)
FORMAT ('O', 'TOTAL POWER INPUT =',
WATT')
F7.1, '
WRITE(8,18)
1111 READ(N,1) TIN, PCOND, TROOM, TITLE, IP
IF(TIN .LE. 0.0) GO TO 999
READ (5, 2) (QW(I), I = 1, 7)
READ (5,3) N, NP, ISUBC, LOSS, KITER, ERROR, ERA
WRITF(6,4) TITLF, TIN, TROOM, PCOND, N, NP, ERROR, EBB
WRITF(8,41 TIT(E, TIN, TROOM, PCONO, N, NP, FRROR, EBB
WRITE(8,19)
C
C
POWER INPUT **********************
C
QTOT = 0.0
DO 100 J = 1, 7
Q(J) = 3.412 * QW(J)
HEATFL (J) = QW(J) / 0.04255
QTOT = Q(J) +
100 CONTINUE
QTOTW = OTOT / 3.412
QWAV WRITE (6, 5) QW, HEATFL
WRITE (6, 6) QTOTW
DISTRIBUTION'
READING AND PRINTING
QTOT
QTOTW/7.0
INPUT
INFORMATION
**********
SIMMARY
SUMMARY
SUMMARY
BASIC CALCULATIONS **
C
C
111
C
C
D = 0.43 / 12.0
ROLIN = ROLTF(TINI
HIN = HLTF(TIN)
PEX = 0.4912 * PCOND
HLEX = HLF(PEX)
HFGFX = HFGF(PFX)
READ(5,2) VIN, PINEST
IF(VIN .LE. 0.01 GO TO 1111
KCOMP = 0
G = 3600.0 * VIN * ROLIN
W = 0.00100B5 * G
WRITE (6,13) VIN, G, W
C
200
201
DO 119 1 = 1,7
QNET(I) = Q(1)
QNETW(I) = QW(I)
CONTINUE
QTOTN - QTOT
PIN = PINEST + PEX
PNVG = PIN
ITS = 1
C
C
C
C
C
C
C
C
C
C
C
C
ZBB = 0.0
ZIRIN = 111.629
7D = 0.0
ZDIN = 111.625
DHTOT = QTOTN / W
HEX = HIN + DHTOT
XEX = (HEX - HLEX) / HFGEX
IF(XEX .GT. 1.0) GO TO 908
C
C
C
XD = 1.OE-30
XEXTR = XEX
IF (ISUBC)121,121,122
CALL SUBBOL (PNVG, ITS, OTD, XD)
XIN = (HIN - HLF(PINI)/HFGF(PIN)
IF(XD .LE. XIN) XD = XIN
IF(XD .LE. XIN) ZD = 111.675
IF((KCOMP+
.GF. KITER) .OR. (IP .GT. 0))
(6,16) PNVG, DTD, X
IF(XO .GE.0.0) GO TO 121
IF((XEX.GT.XD).AND.{XEX/XD.GT.-25.0))
1
XFXTR = XEX - X0*EXP(XEX/XD
IF(XFXTR .GT. 1.0) GO TO 998
IF((KCOMP+1 .GF. KITER) .OR. (IP .GT. 0))
C
C
C
C
C
C
UNHEATED EXIT
LENGTH (3.129
. .
.
OVER
(ISUBC .LE. 0) CALL ROBDRY(ZBB,TB,TBBTR,PBB,0,EB)
= DPFR
DPGREX = DOPGR
DPACEX = OPAC
DPTOTX = DPTOT
PNETEX = PI - .2604* ROLIN / 144.0 - PEX
.....
......
.
P014T 5 PRINTOUT
IFI)KCCMP+1 .00. KITER) .0P. (i1 .GT. 0))
IWRITE (6,10) DPFR, DPGR. DPAC, DPTT, XI, OITR, ALFA,TBI,TBITR,PI,
PNETEX, DPFREX, DOPGRFX, OPACEX, DPTOTX
2
IF
INCHES)
C
DPFREX
C
HEATEC SECTIONS
C
***********
*
*
. .
. .
C
1WRITE
121
.
THE BOILING BOUNDARY IS DETERMINED SY BORDRY ONLY IF NO SU3COOLED
BOILING IS ASSUMED. WITH SUBCOOLFD BOILING, THF TRANSITIlN
THE SCILING ROUNDARY IS SMOOTH AND IT IS NO 4ORE NECESSARY TO
DFTERMINE ITS POSITION ACCURATELY
210
ESTIMATE POINT OF NET VAPOR GENEARATION, USING LFVY'S MODEL (ONLY
IF ISUBC SWITCH IS POSITIVE).
NET VAPOR GENEARATION IS AT FIRST
ASSUMED TO TAKE PLACE IN FIRST TEST SECTION.
122
. .
1WRITE
C
112
. .
DEX
CALL
C
START CALCULATIONS WITH GROSS POWFR INPUT *********
119
RE = )-1.662F-11)*G*G/ROL
ALFA = 0.0
ALFAEX = ALFA
. . . . . . . . .
EXIT PRINTOUT
IF()KCCMP+1 .GF. KITER) .OR. (1P .GT. 0))
AEX, XEXTR, ALFA, TIT, TRETR,
1WRITE (6.)
S))BLIQ(ISUIBC)
GO TO 210
CASE OF POSITIVE EXIT QUALITY . . . . . . . .
TBE = TSATF(PEX)
ROL = ROLF(PEX)
MUL = MULFIPEX)
ROG = ROGF(PEX)
MUG = MUGFfPEX)
XTT = XTTF(XEXTR)
ALFA = ALFAF)XTT)
ALFAEX = ALEA
BE = RF(XEXTR)
TBETR = TF(HEX - XFXTR * HFGEX)
. . . . . .
EXIT PRINTOUT
IF((KCCMP+1 .GF. KITER) .R. (IP .GT. 0))
(6.8) XEX, XEXTR, ALEA, TBE, TBETR, PFX
RELPEX = RFLOF(XEX)
IF (RELPFX .LT. 2000.0) WRITE (6,9) RFLPEX
CALL MARNELISJBC)
IF(XI .GE. 0.0) GO TO 210
DL = (15.5/12.0) / N
L =7 * N
. .
VALUES AT POINT 8
H(1) = HI
X(1) = XI
XTRIE) = XITR
P(1) = PI
= TBI
T(I)
TTR(1) = TBITR
AII) = ALEA
SUMFR(1) = OPFREX
SU4GR(1) = CPGREX
SUMACI1) = DPACEX
SUTOT(I1) = OPTOTX
PNET(1) = PNETEX
C
-
1.0)
WRITE(6,7)
********
DL = 0.2604
DH = C.0
XE = XEXTR
HE = HEX
PE = PREX
QTS = 0.0
IF (XEX .GT. 0.0) GO T1 200
CASE OF NEGATIVE EXIT QUALITY
TBE = TF(HEX)
TBETR = TBE
ROL = ROLTF(TBE)
MUL = MULTF(TBE)
IF(XEXTR .GT. 0.0) GO TO 201
IN THE CASE OF SUBCOOLED BOILING AT THE EXIT MARNEL IS CALLED AND,
FOR SIMPLICITY, SOME PROPERTIES ARE FVALUATED AT THE SATURATION
TEMPERATURE INSTEAD OF THE BULK TEMPFRATURE
.
. .
C
300
DO 800 K = 1,L
M = 7 - (K-1)/N
QTS = ONFTIM)
OH = QNET(M)/ (W * N)
HE
H(K)
XE = XTR(K)
PE = (K)
BE = RI
TBE = TTR(K)
IF (XTK) .GT. 0.0) GO TO 300
CALL SURLIQ(ISUC)
GO TO 310
CALL MARNEL(ISUBC)
TFXI .nF. 0.01
r0 TO 310
.
. .
. .
.
. .
. .
.
. .
.
.
310
C
C
C
C
331
332
IF IISUBC .LE. 0) CALL BDRDRY(ZBBo,TRR,TBRTR,PBB,R,EBB)
H(K+1) - HI
X(K+1)
= XI
XTRIK+1) - XITR
PIK+1
= PI
T(K+1 ) - TRI
TTR(K+1) = TBITR
A(K+1)
= ALFA
FR(K) x OPFR
GR(K) = DPGR
AC(K) = DPAC
TOT(K) = DPTOT
SUMFR(K+1l - SUMFR(K) + FR(K)
SUMGR(K+1I = SUMGRI(K) + GR(K)
SUMAC(K+1) = SUMACK) + AC(K)
SUMTOT(K+1) = SUMTOT(K) + TOT(K)
PNET(K+1I) = P(K+1) - ROLIN * (.7604+ K * DLI/144.0 - PEX
C
C
C
902
DETERMINE THE POSITION OF THF BOILING ROUNDARYBY LINEAR INTERPOLA
TION (ONLY WITH SUBCOOLED BOILING, WHEN BORDRY IS NOT USED)
C
IF(ISUBC)330.330,331
IF ( X(K+1) * X(K)) 332,330,330
PROP = X(KI / (XI(K - X(K+1))
ZBB = 3.125 + 12.0 * DL * (PROP + K
TBB = T(KI + (T(K+1)-T(K)i * PROP
TBBTR = TTR(K) + (TTR(K+1) - TTR(K)}
PBRB
P1K) + (P(K+1)-PIK)) * PROP
C
C
C
C
811
812
-
TRANSFER TO SURBOL
808
330
320
800
C
r
C
809
* PROP
INTERPOLATION
IF (XTR(K+1)*XTR(KI) 320,800,800
XDP = XD
IF(XD .fE. 0.0) XDP = 0.0
PROP = (XDP- X(K)) / (X(K+1) - X(K))
ZD = 3.125 + 12.0 * DL * (PROP + K - 1.0)
TO = T(K) + (T(K+1l) - TfK)) * PROP
TOTR = TTR(K) + ITTR(K+1) - TTR(K)) * PROP
PD = P(K) + (P(K+1l - P(K)) * PROP
KNETVG = M
CONTINUE
CHECK THE SUBCOOLED INLET CONDITION
AT SELECTED POINTS,
FRICTION, GRAVITY,
BFTWEEN PRINTOUT POINTS
PRINTCUT
CALCULATE
901
900
997
998
999
*********
DNPT)F9N),
QWAV, PEX, XEX,
SUMTOT(FIN)
KCOMP - KCCMP + 1
IF(KITER - KCOMP) 111,111,811
TRANSFER TO LOSSES BULK TEMPERATURES AND QUALITIES
LIMITS (ARRAYS ORDERED FROM INLET TO EXIT)
SFCTION
AT TEST
IF (LOSS) 112,112,812
00 S1C I = 1,8
TTR(L+1-(I-l)*N)
TBSThI)
XST(I) * X(L+I-(I-1)*N)
CALL LOSSESITBST,XST,QNETW,G,TROOM,0LOSW)
QTOTN - 0.0
00 820 1 = 1,7
QNETW (1)
QW(I) - QLOSWII)
= 3.412 * QNFTWII)
QNET(I) + QTOTN
QTOTN
GO TO 112
WRITE (6.15) TSATIN
GO TO 111
WRITE(6,991 XEX
GO TO 111
STOP
END
SUBROUTINE SUBLIQ(ISUBC)
******
SPACED NP STFPS
ACCELERATION AND TOTAL PRESSURE
C
WRITE(8,20) VIN. W,
VAPOR GENERATION
C
C
C
C
C
C
C
C
C
C
C
C
OR1
IF((KCOMP+I .GE. KITER) .OR. (IP .GT. 0)) GO TO 901
GO TO 902
LP = L / NP
1,LP
DO 900 1
INI = NP * I1-1 + 1
FIN = NP * I + I
SFR = SIIMFR(FIN) - SUMFP(INI)
SGR = SUMGR(FIN) - SUMGR(INI)
SAC = SI)MACIFINI - SUMAC(INI)
STOT = SUMTOT(FINI - SUMTOT(INI)
(POINT = 7 - (I - ll/(N/NP)
Z = 3.125 + I NP * I) * DL * 12.0
WRITE(6,11) SER, SGR, SAC, STOT, (POINT, Z, X(FINI, XTR(FIN),
A(FIN), TIFIN), TTR(FIN), P(FINI, PNETIFIN), SUMFR(FIN),
I
SUMTOT(FIN)
SUMGP(FIN), SUMAC(FIN),
2
CONTINUE
I
POINT OF NET
IF (ZD .EQ. 0.0) GO TO 809
7DIN = 111.625 - ZD
KITER) .OR. (IP .GT. 0))
(6,171 ZD, ZOIN, KNETVG, TO, TDTR, PD
PD
PNVG
ITS - KNETVG
PIN = P(L+1)
QNET(I)
520
TSATIN = TSATF(P(L+1))
IF(TSATIN .LT. TIN) GO TO 997
C
C
C
C
C
*******
IF(KCOMP+1.GE.
1WRITE
1.01
OETERMINF TEE POINT OF NET VAPOR GENERATION BY LINEAR
CONDITIONS AT BOILING BOU)NDARY
1WRITF
C
C
C
R10
C
C
C
DETERMINE
IF(ZBS .EQ. 0.0) GO TO 809
ZBBIN - 111.625 - ZEN
TS11B4B = TBB - TIN
.GT. 0))
IF((KCCMP+1 .GE. KITER) .OR. (IP
(6,14) Z95, ZBIN, TBB, TRRTR, PB, TSUBBE
XEXTR, ALFAFX,
PIFIN),
PRESSURE DRCP CALCULATION FOR HEATED BUT SIBCOOLFD LIQUID. THE
CALCULATIONS ARE EXECUTED WITH PROPERTIES EVALUATED AT THE AVFRAGE
BULK SFGMENT TEMPERATURE, TBAV AND A CORRECTION IS MADE F3R THV
NON-ISOTHERMAL STATE
RE,
TBE, XD
0, DL, G, DH, QTS, HE, PE,
INPUT VARIABLES
HI, XI, PI, BI, TBl, ALFA, DPFR, OPGR,
OUTPUT VARIABLES
OPTOT, XITR, TBITR
DPEC,
MUL,
MUG, MUGF, KLF
REAL MULB, MULW, MULTF,
COMMON 0, DL, ERROR, G, OH, QTS, HE, HI, XE, XI, PE, PI,
BE, BI, TBE, TRI, ALFA, OPER, DPGR, OPAC, DPTOT, XITR,
TBITR
2
ROL, ROG, MUL, MUG
COMMON /FLUID/
I
XD,
C
C
PRESSURE
DROP FUNCTIONS
*********************
C
OPGRF(ALFA) = -DL*(fROG-ROL)*ALFA + ROL)/144.0
BF(X) = (-1.662E-11l)*G*G*((1.3-X)**2/I().D-ALFA)*ROLI+X**2/(ALFA
*ROG))
1
SUMMARY
C
CALCULATE TEMPERATURES . . . . . .
HI = HF - OH
TRI = TF(HI)
TBAV = 0.5 * (TRF + TBI)
MULB = MULTF(TRAV)
DTFILM = 282.-0*TS*MULB**0.4/(G*0.
TWAV = TBAV + DTFILM
. .
. .
. .
. . .
. .
.
- -
. .
.
- - .
1
2
*KLFITBAV)**0.6)
XD,
PRESSURE DROP FUNCTIONS
C
C
C
CALCULATE PRESSURE DROP
.
. .
..
. .
. .
. .
..
.
..
.
RELPF(X) = G*(1.0-X)*D/MUL
DPLF(X) = (-3.3256E-11)*(DL/D)*FF(RELPF(X))*G*G*(1.0-X)**2/RJL
= -DL*((ROG-ROIL)*ALFA + ROL)/144.0
RF(X) = (-1.6628E-11)*G*G*((1.0-X)**2/((1.0-ALFA)*ROL)+X**2/(ALFA
E
*OG))
MULW = MULTF(TWAV)
ROL = ROLTF(TRAV)
REL = G * D / MULA
DPFR = (-3.3256E-11)*()L/D)*FF(REL)*G*G*((MULW/MULBI**0.14)/R9)L
OPGR = - DL * ROL / 144.0
IF(IISURC .LF. 0) .OR. (XD .GF. 0.01) GO TO I
OPGRF(ALFA)
C
C
C
C
TO AVOID ITERATION THE PROPERTIES ARE EVALIPATFD AT APPROXIMATF
PRESSURFS ANC TEMPERATURES, PAVrST, PIEST
DPEST = OPFR + DPGR
PAVEST = PE - 0.5 * OPEST
PIEST = PE - OPEST
HFGF{PIESTI
XI =(HI - HLF(PIEST))/
GO TO 1
XD) .OR. (XI/XD .LE. -25.0))
IF((XI .LE.
XITR = XI - XD * EXP(XI /XD -1.0)
TRITR = TFIHI - XITR * HFGF(PIFST)I
PSATAV = PSATF(TBAV)
ROG = ROGFIPSATAV)
MUL = MULB
MUG = MUGFIPSATAV)
XTT = XTTF(O.5 * (XE + XITR))
ALFA = ALFAF(XTT)
DPGR = DPGRF(ALFA)
C
C
CALCULATE NOW FOR THE
PSATI = PSATF(TRITR)
ROL = ROLTF(TBITRI
ROG = ROGF(PSATII
INLET
. .
. .
.
. . .
. .
. .
. .
.
. .
1
.
.
MUL= MULTF(TRITR)
1
4
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
REAL MULE, MUGE, MUL, MUG
COMMON D, DL, ERROR, G, OH, OTS, HE, HI, XF, XI, PE, PI,
BE, RI, TRE, TBI, ALFA, DPFA, OPGR, OPAC, DPTOT, XITR,
TRITR
COMMON /FLUID/ ROL, ROG, MUL, MUG
COMMON /NNN/ N
SUBROUTINE
__ _ _CETTE
C
C
C
C
C
IN THIS SECCND MODIFIED VERSION -F MARNFL THE SEGMENT AROUND THE
BOILING BOIINDARY IS SURDIVIDED (I INFARLY WITH X} INTO A BOILING
AND A NCN BOILING REGION AND THE PRESSURE
CALCULATED SEPARATELY
AND ADDED
DROPS
ID
MUG = MUGF(PSATIl
XTT = XTTF(XITRI
ALFA = ALFAF(XTT)
AI = BF(XITRI
GO TO 4
ROL = ROLTF(TBI)
BI = (-1.6628E-11)*G*G/ROL
XITR - -1.DE-20
ALFA = 0.0
ROG = 0.0
TBITR = TBI
OPAC = BE - BI
DPTOT = DPFR + DPGR + OPAC
PTOT
PI = PE XI = (HI - HLF(PI))/HFGF(PI)
RETURN
END
i
12
C
MARNEL(ISURC)
9
SUBROUTINE EST DEDIEE A SU7ANNE
6
PARNEL
CALCULATES THE PRESSURE DROP ACCROSS A SHORT
SUBROUTINE
SEGMENT DL ACCORDING TO THE LOCKHART-MARTINELLI MODEL FOR GIVEN EXIT
AND EXIT QUALITY, XE. THE CALCULATION, EXCrPT
DRESSURE, PE(PSIA),
FOR THE ACCELERATION TERM IS DONE FOR THE AVERAGE PRESSURE, PAV
AND THE AVERAGE QUALITY, XAV.
22
30
THE CALCULATION IS ABANDONED IF NO CONVERGENCE IS OBTAINED WITHIN
30 ITEPATICNS AND AVERAGE VALUES ARE USED FOR SUBSEQUENT CALCULATI INS
0, DL, ERROR, G, OH, HE, XE,
INPUT VARIABLES
TRI, ALFA,
HI, XI, PI, BI,
OUTPUT VARIABLES
TRITR
DPTOT, XITR,
PE,
DPFR,
RE, XD
DPGR,
IF(XITR) 10,11,11
XAVR = 0.5 * XF
DLR = XF / (XE - XITR)
XTT = XTTF(XAVR)
+ (1.0-DIR) R DPLF(0.0)
OPFR = DLB * FI2LF(XTT) * DPLF(XAVB)
ALFA = DLB * AFAF(XTT)
GO TO 1?
XTT = XTTF(XAVI
DPFR = FI2tF(XTT) * OPLF(XAV)
ALFA = ALFAF(XTT)
DPGR = DPGRF(ALFA)
CALCULATE NOW FOR INLET PRESSURE. . . . . . . . . . . . . . . . .
ROL = ROLF(PI)
ROG = ROGF(DI)
IF (XITR .LF. 1.DE-30) GO TO 5
MUL = MULF(PI)
MUG = MUGF(PI)
XTT = XTTF(XITR)
ALFA = ALFAF(XTT)
BI = BF(XITR)
GO TO 6
RI = (-1.6628E-11)*G*G/RL
ALFA = 0.0
XITR = -1.0E-20
OPAC = BE - RI
OPTOT = DPER + DPGR + OPAC
PIFIN = PE - OPTOT
IF(NOITER-30) 21,22,22
WRITE(6,30) PI, PIFIN
FORMAT('D********
CONVERGENCE DIFFICULTY: AFTER 30 ITERATI'
I
,IONS INLET PRESSURE VARIES
F8.3, 1 AND', FR.3,
2
1
PSIA'
/
20X, 'AVERAGE VALUE WAS USED FUR SURSEQUENT'
MARNEL
3
21
DPAC,
NOITFR = 0
CALCULATE FOR THE AVERAGE PRFSSURBE . . . .
HI = HE - OH
PIFIN = PE
PI = PIFIN
NOITER = NnITER + 1
XI = (HI - HLFPI )) / HFGF(PI)
XITR = XI
IF((ISURC .GT. 0) .AND. (XI .GT. XD) .ANO. (X .LT. 0.0) .ANr.
XITR = XI - XD * FEXP(XTI/XD-1.01
1
(XI/XD .GT. -25.0))
XAV = 0.5 * (XF + XITR)
PAV = 0.5 * (PF + PI)
ROL = ROLF(PAVI
ROG = ROGF(PAV)
MUL = MULF(PAVI
MUG = MUGF(PAV)
,
'
CALCIULATIONS **I*******
* (PI
+ PIFIN)
RETWEEN',
PI = 0.5
GO TO 23
ABSERR = ARS(PIFIN - PI)
IF((ABSERR/ABS(DPTDT)).GT.ERRUR).AND.(ARSERR.GT.(ERROR/N)))
GO TO I
1
PI
23
=
PIFIN
IF( XI .GE. 0.0) TRI = TSATF(PI)
IFIXI .LT. 0.0) TRI = TFHI)
TBITR = TBI
IF(ISUBC .GT. 0.0) TBITR = TFIHI-XITR*HFGF(PI))
RETURN
F N)
SUBROUTINE LCSSES(T,X,W,G,TROM,0LOSW)
C
C
C
C
C
C
C
SUBROUTINE LOSSES CALCULATFS THE HEAT LOSSES FROM THE TEST SPCTIONS
USING STANDARD CONVECTIVE AND RADIATIVE HEAT TRANSFER CORRELATIONS.
= 0.527 IN, TEST SFCTION FFFECTIVE LENGTH = 14 IN
TEST SECTION
THE ARRAYS ARE ORDERFD FROM INLET TO
O.D.
C
C
C
C
C
C
C
C
C
C
C
---------
OF
THE B]ILING
SUBROUTINE BOBORY DETERMINES THE EXACT LOCATION
AOUNDARY AND THE PRESSURE DROP IN THE SEGMENT CONTAINING IT USING
BOTH MARNEL AND SUBL10.
INPUT VARIABLES : DL, OH, XE, XI, EBB, K PIUS INPUT VARIABLES OF
MARNEL AND SUBLIQ
OUTPUT VARIARLFS : ZBB, TBR, TBATR, PR9, DPFR, DPGR, DPAC, OPT2T,
PLUS OUTPUT VARIABLES OF SUBLID
COMMON 0, DL, ERROR, G, OH, OTS, HF, HI, XF, XI, PE, PI,
BE, RI, TRE, TBI, ALFA, OPFR, DPGR, OPAC, DATOT, XITR, XD,
TRITR
I
2
C
C
CONSERVE STANDARD VALUES OF DL AND DH . . . . . . . . . . . . . .
DLSTD = DL
OHSTD = OH
BOUNDARY. . . .
DETERMINE BY ITERATION THE LOCATION OF THE
DL = DLSTD
DOL = 0.5 * DOL
0L = DL + SIGN(DDL,XII
OH = (DL / DLSTDI * DHSTD
CALL MARNELI0)
CONVERGENCE IS ACHIEVED WHFN XI IS SLIGHTLY NEGATIVE (TO MAKE SORE
THE ACCELERATION TERM IS CORRECT) AND THF ERROR IN LENGTH SMALLER
THAN EBR
.LT.EBR) .AND. (X1 .LT. 0.01) GO TO ?
GO TO 1
HE = HI
PBB = PI
T8B = T8I
TB3TR = TBI
PRE = PI
BE = BI
TBE = TBI
OPFR1 = DPFR
OPGR1 = OPGR
TPACE = OPaC
OPTOT= DPTOT
DL = CLSTD - DL
DH = DHSTO - OH
CALCULATE LOCATION OF BOILING BOUNDARY . . . . . . . - - . . - - ZBB = 3.12R + K*DLSTD*12.0 - 0L*12.0
CALL SUBLIQ(0)
- . . .
SUM INCREMENTAL PRESSURE DROPS FROM MARNEL AND SUBLIQ
nPFR = DPFR + OPFR1
DPGR1
+
=
OPGR
DPGR
OPAC = OPAC + DPAC1
OPTOT = DPTOT + DPTOTI
RESTORE STANDARD VALUES OF DL AND OH . . . . . . . . . . - . . - .
DL = DLSTD
OH = RHSTD
RET1IRN
END
BOILING
C
I
C
C
C
(F((DDL
2
C
C
C
EXIT
REAL MULTF, KW, KLF
DIMENSION T(B), X(), QW(7),
SUBROUTINE BCBDRY(ZBR,TBS,TBBTR,PBBK,FBI)
C
C
C
CALCULATE AVERAGE BULK
1,7
DO 9 J
11(j) =J
TAV = 0.5 * (T(J)
C
C
C
CALC'JLATF FIL
C
I
C
C
2
C
C
C
M
+
QLOSW(7),
11(7)
TEMPERATURES . . . . . . . . . . . . . . .
T(J+1))
TEMPEPATURE DROP,
TF
. .
. .
.
. .
.
. .
.
. .
.
.
IF(X(J+11) 1,1,2
0.023 * RE**O.R * PR**0.4
NO
WITHOUT BOILING:
DTF = 1060.0 * QW(J) R MULTF(TAV) **0.4 /(r**0.B*KLF(TAV)**0.6)
GO TO 3
WITH BOILING A HEAT TRANSFFR COEFFICIENT Or 1000 RTJ/HR-rT2-F
IS USED
OTF = 0.026 * OW(J)
CALCULATE WALL TEMPERATURE DROP, DTW FOR AVERAGE WALL TEMPERATURE
3
4
C
C
5
C
C
C
DTWFI =0.0
DTW = OTWFI
HEAT CCNDUCTIVITY OF THE PYREX GLASS WALL :
RTI/HR-FT-F
KW = 0.67 + 0.0005 * (T(F) - 100.0)
KW = 0.67 + 0.0005 * (TAY + DTF + 0.5 * DTW - 100.0)
DTWFI = 0.0952 * QW(J) / KW
IF (ABSIDTW - DTWFI) - 0.1) 5,5,4
DTW = DTWFI
CALCULATE OUTSIDE
WALL TFMPERATURE
.
.
.
. .
. .
. .
.
. .
.
. .
.
TOUT = TAV + OTF + DTW
C
C
C
C
C
9
CALCULATE HEAT LOSSES IN WATTS USING THE FORMULA
FOR CONVECTIVE LOSSFS
H = 0.29 * (DT/L)**O.25
FOR RADIATIVE LOSSES . . . .
AND AN EMISSIVITY OF 0.q5
. .
. .
C
10
.
DLOSW(J) = 0.0132*(TOUT-TROOMI*(ARS(TOUT-TRoAM))**0.25 + 0.00768*
f({TOUT+459.6)*0.01)**4 - ((TROOM+459.6)*0.01)**4)
1
WRITE 16,10) G, TROOM, II, T, X, QW, QLOSW
FORMAT I/// 20X, 58HHEAT LOSSES CALCULATFD FROM THE TEMP. DISTR.
, 5X, 4H(G = , F10.1. rX, 7HTROOM =, F6.1, OH I
1IELOW
/
, 7I10
22HOTEST SECTION NUMBER
I
/
, SF10.1
17HOTEMPERATOEF IF)
2
, BF10.3 /
17HOEQUIL. QUALITY
3
/
, 7F10.1
22HOPOWER (WATT)
4
, 7F10.1
22HOPOWER LOST (WATT)
5
RETURN
END
(P,ITS,DTD,VXD)
SUBROUTINE SUBROL
C
C
C
C
C
C
C
C
C
-
THIS VERSION OF SUBROUTINF SUBBOL CALCULATES THF POINT OF NET
VAPOR GENERATION ACCORDING TO STAUB'S MODEL.
0,G
P, ITS, QNET(7),
INPUT VARIABLES :
P : PRESSURE AT POINT OF NET VAPOR GENFRATION (PSIAI
ITS : TEST SECTION NUMBER (SELECTS THE PROPER HEAT FLUX VALUEI
INLET TO EXIT
: NET DOWERDISTRIBUTION (RTU/HR),
ONET)
AND
QNETARE
ORDERED FROM INLET
C
(ITS
C
C
C
C
C
C
n
C
SURROL REQUIRES THE USE OF FUNCTIONS
DIAMETER
TO EXIT)
(FT)
MASS FLUX (LBM/HR-FT2)
G
OUTPUT VARIABLES : DTD, XD
TSAT - TRULK AT POINT OF NFT VAPOR GENFRATION (F)
OTD
XD : NEGATIVE EQUILIBRIUM QUALITY AT POINT OF NET VAPOR GENERATION
TSATF(P)
C
C
, ROGF(P)
0,
ROLF(P),
MULF(P),
SIGMF(T),
AND KLF(T)
REAL MUL, MULF, KLF, KL
DUM2) , G /SUBSUB/
COMMON
SUBROUTINE SURROL (P,ITS,DTD,XDi
-____---__-________---__-____-_-)
ST AUB
-----
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SUBROUTINE S'BROL CALCULATFS THE POINT OF NET VAPOR GENERATION
USING LEVY'S MODEL.
P, ITS, QNET(7), 0, G
INPUT VARIABLES :
P : PRESSURE AT POINT OF NET VAPOR GENERATION IPSIA)
ITS : TEST SECTION NUMBER (SELECTS THE PROPER HEAT FLUX VALUE)
INLET TO EXIT
: NET POWER DISTRIBUTION (BTU/HRI,
QNET17
ARE ORDERED FROM INLFT TO EXIT)
(ITS AND
T) DIAMETER (FTI
MASS FLUX (LBM/HR-FT21
G
OUTPUT VARIABLES : DTT, XD
NET VAPOR GENFRATION IF)
TSAT - TRULK AT POINT
DTD
AT POINT OF NET VAPOR GENFRATION
XO
NEGATIVE EQUILIBRIUM
QNET
OF
QUALITY
SUBBOL REQUIRES THE USF OF
TSATFIPI AND KLF(T)
C
C
3
I
2
RELATION
THE FRICTICN FACTOR IS OBTAINED FORM AN EMPIRICAL
0
D33)
F = 0.0055*(1.0+(2.OE+04*ROUGH+1.OE+06/REL)**0.
RP = F * B
DO = (-BP+SQRT(BP*BP+A))/A
IF(ABS(DD-ODSAVE)/DD-0.001)2,2,1
ROUGH = 0.5*DD/D
DDSAVE = O
GO TO 3
0.5*DD*G*SQRT(F/8.0/MUL
YD
YCL = (D/DID)*Y
Q = 86.1 * QNET(ITS) / (G * SORTIF)
DTD = Q*(PR*(5.0-YD)+5.0*ALOG(1.0+5.0*PR)+
IF(YD .LE. 5.0)
2.5*((A1OG(YCL/30.0))/(1.0-30.0/YCL)-l.0))
1
DTD = 0*(5.0*ALOG((1.
30.0))
.GT. 5.0) .AND. IYD .LE.
+5.0*PRI/11.0+PR*(YD/R.0-1.01))+2.5*(ALOGIYCL/30.1/(1.0-30.0
1
/YCLI-1.0l)
2
OTD = ?.5Q*iALOG(YCL/YD)/(1.O-YO/YCLI-1.0
IF(YD .GT. 30.0)
)*(1.O-YD/YCL)
1
XD = -0.00364 * DTD
IF((YD
C
7
WRITE(6,7) FRREL,F,DD,YD,DTD,XD
FORMAT('OPOINT OF NET VAPOR GENERATION ACClRDING TO STAUB''S MODEL
F6.3, 4X, 'RE =', F10.1, 4X, 'F =', F7.4,
=',
' F(BETA)
1 '/
F7.2
4X, 'ODD =', F9.6, I FT', 4X, 'YT =', FR.2, TX, *DT2 =',
2
'XD =' , F8.4)
, ' F' , 4X,
3
RETURN
END
MIILFIP). SIGMF(TI,
ROL = ROLF(P)
MUL = MULF(P)
T = TSATFI)
SIGMA = SIGMF(TI
KL = KLF(T)
PR = 0.225 * MUL f KL
REL = G * 0 / MUL
F = 0.0055 * (1.0 + (2.0 + 1.OE 06/REL)**0.33331
THE FRICTION FACTOR IS OBTAINED FORM AN EMPIRICAL RElATION FOR
A RELATIVE ROIGHNESS OF E-04
0 = 86.1 * QNETITS) / (G * SQRT(FI))
YBPLUS = 308.0 * SQRT(SIGMA * 0 * ROL) / M'IL
DTFILM = 2R2.0*QNET(ITS)*MUL**0.4/(G**O.R*KL**0.6I
* YRPIUS
IF (YBPLUS .LE. 5.0) OTD = DTFILM - Q *
IF ((YBPLUS .GT. 5.0) .AND. (YBPLUS .LE. 30.0)) DTD =
DTFILM - 5.0 * Q * (PR + ALOG(1.0 + PR * (YRPLUS/5.0 -1.01))
1
IF (YBPLUS .GT. 30.0) DTD =
2TFILM - 5.0 * Q * (PR+ALOG(1.0+5.0*PR)+0.5*ALOG(YBPLUS/33.0)I
A
XD = -0.10364 * DTD
= NULF(P)
TSATF(PI
SIGMA = SIGMF(T)
KL = KLF(T)
PR = 0.225 * MUL / KL
REL = G * 0 / MUL
A = (ROL-ROG)/112.0*SIGMA*FB)
B = G*G/(64.0*32.17*ROL*SIGMA*FR*3600.0*3600.0)
ROIJGH = 0.0
DOSAVE = 0.0
ROLF(P),
C
FR = 0.030
ROL = ROLF(P)
ROG = ROGF(P)
MUL
T =
FUNCTIONS
REAL MUL, MULF, KLF, KL
COMMON D, DUMI?), G /SUBSUB/ QNET(7)
QNETI(7
C
IEVY
C
C
PR
C
I
E
2
3
WRITE(6,1I REL, F, YRPLUS, OTFILM, DTD, XD
FORMAT('OPOINT OF NET VAPOR GENERATION ACCORDING TO LEVY''S MODEL'
= ',
FM.?, NX,
/
I RE =', F10.1, 5X, 'F =-, r7.4, 5X, IYRP
'DTFILM = I, F7.2, * F', 15X, 'OTT =', F7.2, 10X, 'XD =',
F8.41
RETf)RN
END
lUS
0=0.431
Appendix B
The Physical Properties of Freon-113
The physical properties of Freon-113'as a function of saturation
pressure or temperature were taken from references [41] and [42].
These
functions were fitted in the least squares sense to polynomials (using
IBM-360 Scientific Subroutine Package subprograms) with an excellent
accuracy.
The resulting FORTRAN FUNCTION subprograms are given in the
following pages.
To evaluate the accuracy of the fits the entire thermo-
dynamic tables in the range
40 - 170*F, with an increment of 10*F,
were reconstructed from these functions ard compared to the original ones.
The resulting r.m.s. errors were between 0.0002 and 0.2 percent.
0
C
C
C
C
FUNCTION TSATFIP)
SATURATION TEMPERATURE (F)
X
= 1.0 / (ALOG(P)
- 13.47)
TSATF = - .360185E 03 - .4069271E
RETURN
END
C
C
C
C
C
C
S
F R E C N - 1 1 3
P R
P E R T I F
-__ -- - - - - _ _ - -- - _-- _-_ - ARGUMENTS IN DFGREES F IT) OR PSIA (P).
C
04*X
+ .117441E
05*X=X
FUNCTION ROLF(P)
I 10010 DENSITY
(LBM/FT3)
X = ALOG(P)
ROLE
=
.1033078F
03 - .2657116E 01*X
I
.3389485E -01*X*X
- .83079RIF -01*X*X*X
RETURN
END
C
FUNCTION ROGF(Pl
VAPOR DENSITY
)LBM/FT3)
X = ALOG(P)
ROGF
= EXP(
- .3267308E 01
1
+ .3519715E-02 *X*X
RETURN
END
C
+ .9181870E
00 *X
REAL FUNCTION MULF(P)
LIQU1D VISCOSITY
(LRM/HR FT)
X = ALOG(P)
MULF
= EXP(
.1064657 F 01 - .3255410 F 00*X
- .1726090 F-02*X*X*X I
+ .6064336 E-02*X*X
1
RETURN
END
REAL FUNCTION MUGFIP)
VAPOR VISCOSITY
(LBM/HR FT)
X = ALOG(P)
MUGF
=
.1967795 E-01 + .7899068 F-02*X
- .5630545 E-02*X*X
+ .2141094 F-02*X*X*R
2
- .3536185 E-03*X*X*X*X
+ .1974262 F-04*X*X*X*X*X
RETURN
END
FUNCTION HLF(P)
LIQUID ENTHALPY
X
= 1.0 /
HLF
+ .7712507
RE TURN
FND
r
C
C
IBTU/BLM)
- 13.54RAR)
= - .3302970 F 02 + .2946777 E 00*X
E 04*X*X
C
C
C
C
FUNCTION OHLF(P)
DHLF IS THE SLOPE OF THE LIQUID SATURATION FNTHALPY VS ARSOLUTE PR
rSSURE (ORTAINED FROM DIRECT DIFFEPENTIATION OF HLF)
UNITS : BTU/LBM-PSIA
X = 1.0 / (ALOG(P) - 13.4888)
OHLF = -(0.2946777 +
* X) * X *X / P
RETURN
END
155A5.014
(BTU/L9M)
FUNCTION PSATF(TF
SATURATION PRESSURE
(PSIA)
T = TF + 459.6
PSATF = 10.0 **(33.0655 - 4330.98/T -9. 2635*ALOGI0(T) +
1
0.0020539 * T)
RE TURN
END
FUNCTICN ROLTF(T)
DENSITY
(LRM/FT3)
ROLTF
=
.1035453 E 03 - .7105255 E-OI*T
.6448694 E-04*T*T
1RETURN
END
lIQUID
REAL FUNCTION MULTFIT)
LIQUID VISCOSITY
(LBM/HR FT)
X = 1.0 /tT + 459.6)
MlLTF
= EXP( - .S483695 E 01
+ .55R921 F 04*X
- .1950534 E 07*X*X
+ .3577018 E 00*X*X*X)
RETURN
END
1
(ALOG(P)
1
FUNCTICN HFGF)P)
LATENT HEAT OF VAPORISATION
HEGF
=
.7058670 E 02 - .9218673 F 00*P
I
+ .4283483 F-OI*P*P
- .1186130
F-32*P*P*P
2
+ .1250690 E-04*P*P*P*P
RETURN
ENQ
C
FUNCTION HLTF(T)
LIQUID ENTHALPY
(BTU/LBM)
HLTE
.8119482 E 01 + .1961794 E 00*T
=
I
+ .1261047 E-03*T*T
RETURN
END
REAL FUNCTICN KLF(T)
LIQUIC THERMAL CONDUCTIVITY
KLF = 0.0475 - 6.7667E-05*T
RETURN
END
(BTU/HR FT F)
FUNCTION SIGMF(T)
SURFACE TENSION
(LRFIFT)
SIGMF =
- 4.45E-06 * IT
RETURN
END
1.27E-03
FUNCTION TF(HL)
SATURATION TEMPERATURE (Fl
TF
= - .4082034
1
- .1141203
E-01*HL*HL
RFTURN
F ND
-
80.0)
E 02
+
.5196681
F 01*HL
NWIhlll I ulN
Appendix C
Auxiliary Functions for Pressure Drop Calculations
This appendix describes the evaluation and use in the computer
programs of the friction factor, and the Lockhart-Martinelli two-phase
friction multiplier
C.1
2
$ t
RJtt
a.
and void fraction
The Friction Factor, f
The Fanning friction factor, f for smooth pipes was obtained from
Ref. [48] and approximated by a polynomial in the least squares sense.
The resulting FORTRAN FUNCTION subprogram is listed at the end of this
appendix,
C.2
and the
accuracy of the fit is given in Table C.l.
r.m.s.
The Lockhart-Martinelli Two-Phase Friction Multiplier,
The values of
2
4t
as a function of
ktt
X
tt
(P2
ktt
are tabulated in Ref. [49].
A least squares fit by a sixth order polynomial gave a satisfactory
accuracy as shown in Table C.l.
It was, however, necessary to extend the
range of the function to very low and very high qualities:
By definition:
= 02ktt
(4k)
dL 2fr
dp2
dL
gtt
(
)
dL
C-2
with
tt
gtt
X=
ttt kt
At x approaching zero,
X
tends towards infinity and
obviously approach the unity.
lim
X *0 tt
=
2
4ptt must
Therefore:
1
tt
x + 0
When
x
approaches the unity,
X
tt
must have the limiting value of 1.
2
lim
X +>
-
tt
tt
tends towards zero and
tt
As
=
gtt /Xtt
2
0g
gtt
now
it follows that
1
X2
tt
x + 1
These limiting cases are reflected in the FORTRAN listing of this
subprogram.
C.3
The Lockhart-Martinelli Void Fraction Correlation
A numerical fit of the void fraction,
was obtained in exactly the same way as for
was arbitrarily set equal to 0.97 for
Xtt
a, as a function of Xtt
2
@Ptt
The value of
values smaller than
(corresponding to qualities a few percent below unity).
end of the range,
for
Xtt
larger than
a
0.07
At the other
100 (x smaller than 0.007)
asymptotic expression, suggested by the homogeneous model, was used.
an
C-3
Table
C.1
Range and Accuracy of the Numerical Fits of the Auxiliary Functions
FORTRAN
FUNCTION
name
Variable
Spacing of
the input
and test pts.
Range
of fit
Relative
rus error*
()
Maximum observed
relative error*
()
Friction factor,
f(Re)
FF(RE)
3500<Re<300,000
9 values per
decade
0.20
0.48 @ Re-5000
Lockhart and Martinelli's friction
multiplier .2 (x
Itt tt
F12LF(XTT)
0.01<X <100
4 values per
decade
0.80
1.53
Lockhart and Martinelli's void
fraction, 0(Xtt)
ALFAF(XIT)
0.07<X tt<100
4 values per
decade
2.01
4.86 @ X
*
using spacing of test points given in this table.
fit better than 1% for
Xtt < 10
a
or
> 0.47.
AUXILIARY FUNCTIONS FOR
PRESSURE
OROP CALCULATIONS
FUNCTION FF(RE)
FANNING FRICTION FACTOR FOR SMOOTH PIPES
X * ALUG(RE)
FF
EXP(
- .1805868 E 00 - .8502377 E 00*X
- .1134451 F-02*X*X*X )
I
+ .4752425 E-01*X*X
RETURN
END
C
FUNCTION XTTF(X)
LOCKHART-MARTINELLI XTT PARAMETER
REAL MUL, MUG
COMMON /FLUID/
ROL, ROG, MUL, MUG
IF(X .GT. 1.0E-301 XTTF =
(((1.0 -X)/X)**0.9)*((ROG/ROL)**0.5)*(MUL/MUG)**O.1
1
IF(X .LE. 1.OF-30)
XTTF = 1.0E 30
RETURN
END
FUNCTION ALFAF(XTTI
LOCKHART-MARTINELLI
IF(XIT
VOID
fRACTION
CPRRELATION
.LT. 0.07) ALFAF = 0.97
10.2661 / XTT
IF(XTT .GF. 100.0) ALFAF
X = ALOG(XTT)
IF((XTT .GE. 0.07) .AND. IXTT .LT. 100.0)) ALFAF
EXP(
- .25435?1 E 00 - .1478533 E 00*X
I
+ .1222962 E-0?*X*X*X
2
- .3270721 E-01*X*X
- .4598919 E-0*X*X*X*X*X
3
+ .4128012 E-03*X*X*X*X
RETURN
END
=
FUNCTION
F2LF(XTT)
C
PRESSURE
DROP MULTIPLIFR
LOCKHART-MARTINELLI TWO-PHASF
1.0 /(XTT*XTT)
IFIXTT
.LT. 0.01) FI2LF
IF(XTT
.GE. 210.0) FI2LF = 1.0
X = ALOG(XTT)
IF((XTT .GE.
.AND. (XTT .LT. ?10.0)) FI2LF =
.1445175 E 01 - .5039395 E 00*X
( EXP(
- .1157590 E-02*X*X*X
+ .578383 E-01*X*X
2
+ .3007357 F-04*X**5
- .4390361 E-O0*X*X*X*X
RETURN
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Appendix E
S
t
THR
S
HOL
T A
U
d
I
LI
T Y
AND
'L X P L R
TR ANS
I
ME NT
I T 10 N
v
S
)A T A
INT
OW - AVERAGE 6RUSS POWER PER TEST LENuTh
TIN - INLET TEMPLRATURL
AVtRAGE MASS FLOW RAT.
Pu - PRESSURE AT STATIUN k,
Pi - PRESSURE AT -TATION 1
DP1EX - PRESSURE ORUP FROM HLATEO INLAT TO EXIT (1 TO tX}
XEX - AVERAGE EXIT OUALITY
DTSUIN - INLET SUdCOOLING* TSATIP1l - TO
DTSUdb
- SUBCOOLING Wil-H RESPECT TO TML OUILIN16 OUNUARY
TbAT(PobI - TO
Q/WHFG - TOTAL HEAT INPUT / MASS FLOW RATE TIMES LATENT
TAU - PERIOD OF THE OSCILLATION
WP/WA - PEAK TO AVERAGE FLOW RATIO
DPRAT - SINOLh-PrIASt TO TOTAL PRESSURE OxOP MATIU (0 TO Lb / U TO EL)
ZRAT - SINGLE-PHASE TO TOTAL LEN6TH RATIO (1 TO do / 1 TO
DHRAT - SINGLE PHASE TO TOTAL ENTHALPY RATIO
DOLDH - DELAY OF ENTHALPY PERTURbATION AT Ubs IN CYCLES (FROM 04U*T-)
ORDER - ORDER OF THE OSCILLATION
W-
HEAT
EX)
COOL FOR NUMOERINu OF THL POINTS
d0
9
100
200
900
SERIES
SERIES
SERIES
SERIES
SERIES
U N
RUN-PT
OW
IWI
TR2
TR2A
DIZA
012A
TR28
T728
TR28
D4
DS
D5
09
D10A
017
016
018A
018A
020
D20
D20
D20
D20
91
91
92
93
92
93
94
294
932
299
298
90
296
293
196
187
98
189
90
82
83
RUN-PT
DIZ
TR3
D4
D6
06
05
09
010A
010A
TR5
D17
017
0184
0ISA
D18A
318A
018A
DISA
016
016
018
018
D18
020
020
020
98
92
917
199
196
294
94
93
284
92
197
182
191
182
93
185
287
280
97
289
290
186
184
90
184
186
N
TIN
PO
(F) (LBN/HRI (PSI)
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
121.5
123.0
123.5
123.2
119.2
118.9
118.3
116.6
103.6
105.8
101.9
91.5
85.3
72.8
62.5
62.3
44.3
44.2
44.8
45.1
45.2
ON
(N)
TIN
(F)
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
125.1
120.0
118.0
115.6
114.0
107.0
103.2
94.0
93.8
93.3
85.4
84.9
62.6
63.0
63.4
64.0
64.1
64.2
72.4
72.4
61.4
62.3
61.8
42.5
44.0
45.1
279.0
419.0
700.0
572.0
168.5
149.0
131.0
285.8
85.0
171.2
165.5
47.9
153.5
103.4
102.0
83.0
113.0
96.6
88.8
75.9
63.5
N
(LB/HRI)
516.0
262.0
281.0,
265.0
211.0
223.5
202.5
194.5
142.8
262.0
214.0
87.0
440.0
300.0
197.0
143.0
122.5
88.0
140.5
126.4
198.8
103.5
119.0
204.0
116.0
81.3
I
-
TRANSITIONS
FIRST OCCURENCE OF USCILLATIONS
FREUUENCY OF NEXT UNSTAbLE POINT WAS USED
THRESHOLD ObTAINED 6Y INTERPOLATION
TRANsITIONi TO FUNDAMENTAL MODE
F 0 R M
P1
DPEX
(PSI)(PSI)
16.54
19.61
21.26
20.43
18.34
18.18
18.05
19.23
17.82
19.52
19.12
17.62
19.80
20.20
20.02
19.37
21.80
21.46
21.12
20.54
20.10
17.51
18.53
20.03
19.28
17.34
17.18
17.06
18.20
16.82
18.50
18.11
16.61
18.77
19.17
18.98
18.34
20.74
20.41
20.07
19.49
19.05
PO
(PSI)
P1
(PSI)
21.10
19.15
19.39
19.38
19.06
19.95
18.86
19.09
18.57
19.99
19.56
17.85
21.35
21.07
19.82
19.18
17.36
16.54
19.70
19.20
20.18
18.61
19.36
21.33
19.61
18.78
19.98
18.13
18.36
18.35
18.05
18.90
17.84
18.06
17.55
18.94
18.52
16.83
20.21
19.98
18.76
18.14
16.32
15.50
18.66
18.17
19.1?
17.57
18.32
20.25
18.55
17.73
2.96
3.98
5.12
4.37
2.64
2.48
2.36
3.46
1.98
3.66
3.66
1.95
4.17
4.43
4.61
3.97
5.84
5.51
5.17
4.59
4.15
DPIEX
(PSI)
H E A T
F L U X
U I
b T RI
XEX OTSUIN DTSUB8 J/WHFG TAU
(F) IF)
(S)
0.152
0.112
0.073
0.084
0.231
0.259
0.292
0.129
0.394
0.177
0.176
0.699
0.135
0.209
0.184
0.268
0.081
0.138
0.174
0.248
0.345
5.6
7.3
11.2
9.4
7.4
7.1
7.4
12.6
21.3
24.7
26.5
32.7
45.6
59.9
69.1
67.4
92.4
91.690.1
88.1
86.7
3.7
4.2
5.0
4.6
5.6
5.6
5.9
8.3
18.3
18.7
20.8
29.9
35.7
50.3
58.4
58.5
77.4
78.6
78.1
77.8
77.9
0.136
0.090
0.054
0.066
0.225
0.254
0.289
0.132
0.446
0.221
0.228
0.790
0.246
0.366
0.371
0.455
0.335
0.392
0.427
0.499
0.597
XEX 0TSUIN DTSU88 Q/WHF
(F
(F)
M
5.17
0.172
3.43
0.298
3.62 '0.271
3.51
0.277
3.21 0.344
4.06 0.799
3.39
0.326
3.40 0.307
2.89
0.446
4.24
0.203
3.92
0.242
2.23
0.756
5.84 -0.014
5.61 0.067
4.39
0.200
3.77
0.347
3.43
0.453
2.61
0.694
3.92 0.381
3.43 0.440
4.53
0.187
2.98
0.541
3.73 0.444
5.35 0.111
3.65
0.398
2.83
0.680
9.5
9.1
11.7
14.1
14.8
24.7
24.9
34.9
33.4
38.2
44.8
40.0
72.6
71.6
67.6
65.1
64.0
61.2
58.3
56.8
71.0
65.0
67.9
92.8
86.3
82.7
6.6
7.3
9.4
11.4
12.5
20.4
21.1
29.6
29.6
31.0
37.6
37.0
49.8
56.4
57.4
57.6
57.4
56.5
51.8
51.0
60.0
59.4
61.3
78.8
78.2
77.0
0.147
0.289
0.269
0.286
0.359
0.339
0.373
0.389
0.530
0.289
0.354
0.870
0.172
0.252
0.384
0.529
0,617
0.659
0.539
0.599
0.380
0.731
0.636
0.371
0.653
0.932
4.20
4.65
4.00
3.94
3.60
3.65
3.80
2.95
8.65
3.20
3.57
7.25
2.64
3.08
3.90
5.85
3.80
3.584.20
3.30
5.80
TAU
(S)
3.00
3.52
3.96
3.81
5.00
5.40
2.74
6.60
6.4C
7.50
7.25
6.15
6.15
2.84
4.68
9.05
8.49
0.00
8.70
2.32
4.70
4.00
3.80
6.00
8 u T
Iu
WP/WA OPRAT
1.10 0.403
0.409
1.16 0.518
1.36 0.485
1.27 0.420
1.34 0.422
1.33 0.425
0.529
6.05 0.627
0.601
0.630
0.612
0.788
0.720
0.772
0.747
1.30 0.858
1.40 0.802
1.20 0.787
1.40 0.767
1.20 0.730
WP/WA
DPRAT
0.333
1.15 0.351
1.21 0.382
0.415
0.414
0.456
1.68 0.503
~
0.597
0.563
1.06 0.631
0.665
0.590
1.051
0.872
0.774
0.694
0.672
0.664
0.620
0.635
0.786
0.684
-.-. 0.644
1.20 0.864
1.90 0.760
2.50 0.718
N
ZRAT DHAAT
0.094
0.161
0.325
0.245
0.067
0.077
0.071
0.719
0.142
0.293
0.314
0.130
0.496
0.467
0.532
0.433
0.773
0.671
0.613
0.522
0.438
0.097
0.165,
0.335
0.252
0.090
0.079
0.073
0.225
0.146
0.301
0.324
0.134
0.510
0.481
0.547
0.446
0.795
0.690
0.631
0.537
0.450
RAT DHRAI
0.15?
0.089
0.122
0.140
0.121
0.210
0.195
0.262
0.192
0.369
0.364
0.145
1.000
0.755
0.504
0.368
0.314
0.222
0.327
0.290
0.532
0.274
0.326
0.709
0.401
0.277
0.162
0.091
0.125
0.144
0.124
0.216
0.201
0.269
0.198
0.380
0.315
0.150
1.000
0.776
0.519
0.379
0.323
0.228
0.337
0.298
0.548
0.262
0.335
0.730
0.412
0.285
DELDM
ORDER CO4MENTS
0.67
0.686
0.732
0.714
0.734
0.7?7
0.725
0.936
0.947
0.914
0.919
0.621
0.784
0.745
0.935
0.018
0.755
0.794
0.719
0.150
0.603
0.256
0.262
0.368
0.344
0.456
0.448
0.455
0.827
0.622
1.719
1.719
1.220
4.009
4.854
4.464
2.981
6.095
0.000
5.565
7.055
4.022
808. HX
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0ELDH
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COMMENTS
0.697
0.719
0.752
0.754
0.831
0.823
0.765
0.865
0.869
0.879
0.910
0.707
0.744
0.427
0.888
0.929
0.801
0.934
0.836
0.913
0.359
0.392
0.424
0.480
0.603 INTERPOL
0.576 TAU ?
TAU-6.60
0.662 INTERPOL
0.717
0.743
TAU ?
0.755
1.233 W ESTEST
0.000 W7ENDOSC
3.006
1.833
0.945 INTERPOL
0.990 INT@RPOL
0.000
0.872 INtERPOL
3.856 INTERPOL
1.881
0.894
0.930
0.936
2.948 THA ?
3.078 TAU CAHN
1.919 TAU CHAN
INTERPOL
W EST
INTERPOL
INTERPOL
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RUN-PT
QW TIN
(W)
(F)
D7
08
011
D22
022
D22
D23
D23
D23
D23
023
023
D23
D23
D21
D21
D21
D21
D21
300 114.0
300 103.7
300
94.0
300
83.6
300 83.4
300 83.4
300 69.8
300
71.0
300
71.1
300
71.2
300
71.3
300
71.6
300
71.7
300
72.6
300 42.0
300
42.4
300 43.3
300 44.2
300 42.5
198
199
192
98
181
82
195
88
89
80
81
82
83
86
91
93
86
88
92
RUN-PT
QW
(WI
W
(LRM/HR)
349.0
312.0
296.0
444.0
275.0
164.5
410.0
189.0
166.0
159.0
154.5
142.0
120.0
84.1
304.0
751.0
156.0
141.0
320.0
Po
(PSI)
N
t
P1
DPlEX
(PSI) (PSI)
21.15 20.09
19.65
20.63 19.57
20.99 19.86
20.37 19.31
19.52 18.49
20.91 19.79
19.72 18.67
19.64 18.60
19.56 18.52
19.50 18.46
19.36 18.32
19.12 18.09
18.32 17.29
20.51 19.41
20.28 19.19
19.68 18.62
19.36 18.30
20.99 19.88
20.70
XEX OTSUIN DTSU88 Q/WiFG
(F)
(F)
5.26
4.91
4.91
5.12
4.57
3. 75
5.06
3.94
3.87
3.79
3.73
3.59
3.36
2.56
4.85
4.63
4.06
3.74
5.32
0.311
0.314
0.301
0.136
0.293
0.570
0.110
0.438
0.522
0.552
0.573
0.639
0.786
1.193
0.116
0.196
0.475
0.555
0.099
W
Po
Pl DPlEX
(LAM/HR) (PSI) (PSI) (PSI)
TIN
IF)
D8 191 400 104.2
011 193 400 94.6
022 82 400 83.5
D22 94 400
83.7
D22
86 400 83.5
023 92 400
70.1
D21 92 400
41.3
470.0
385.0
268.0
420.0
283.0
430.0
440.0
22.17
21.63
20.52
21.66
20.94
21.47
21.27
21.05
20.54
19.46
20.55
19.88
20.34
20.11
S E R
20.9
29.9
39.4
50.6
49.2
46.8
64.2
59.7
59.4
59.0
58.7
58.0
57.7
53.8
90.9
89.9
87.3
85.4
91.8
16.2
24.2
32.8
40.4
41.7
41.2
52.4
5?.6
52.9
52.7
52.5
52.1
51.8
49.6
77.9
78.7
79.0
77.7
78.4
0.274
0.313
0.445
0.241
0.415
0.187
0.085
33.4
41.6
49.6
52.5
50.8
65.5
93.7
S T A
E
27.1
35.0
43.3
44.2
44.3
55.5
79.6
I
TAU
IS)
ZRAT DHR AT
0.417
0.491
0.54?
0.709
0.620
0.584
C.791
0.654
0.629
0.627
0.628
0.675
0.615
0.643
0.866
0.803
0.704
0.712
0.834
1.07
2.75
4.20
1.15
4.60
4.30
4.20
4.50
3.50
4.30
7.50
1.24
1.19
1.80
5.40
1.09
0.690
WP/WA DPRAT
0.322
3.29
0.393 4.50
0.565
9.72
5.10
0.361
0.535 10.40
6.00
0.344 6.30
0.444
0.479
0.536
0.562
0.516
0.657
0.836
3.00
1.06
2.90
1.16
1.05
0.352
t
U I 140
WP/WA DPRAT
0.326
2.83
0.364
3.73
0.384 4.50
0.256
5.60
0.413 11.72
5.10
0.277
6.40
0.601
3.00
0.684
6.24
0.714
2.50
0.735
5.49
0.800 2.28
0.944 4.02
1.350 6.15
0.373 6.80
0.452
7.10
0.727
7.48
0.805
3.61
0.355 6.00
L I T Y
Io
, T k
XEX DTSUIN DTSUBB Q/WHFG TAU
(F)
(F)
(S)
6.31
5.88
4.72
5.81
5.14
5.61
5.55
I ES
u I
F L UX
H L A T
X P ck
DELDH
0.643
0.788
0.765
0.891
0.663
0.933
0.803
0.756
0.719
0.812
0.826
0.841
0.816
0.690
0.396
1.110
0.75?
7.065
1.047
2.655
1.231
3.038
1.839
1.314
1.042
1.047
1.165
2.473
1.175
DELDH
ORDER
COMMENTS
0.769
0.759
0.713
0.779
0.705
0.772
0.832
0.778
0.676
0.412 TAU CHAN
0.677
W 7
0.382 TAU=5.08
0.668 W? 1AU'
0.846
ZRAT DHRAT
DELDH
ORDER COMMENTS
0.703 0.723
0.599
0.5?7 0.542
0.500 0.515
C.470 0.483
0.442 0.454
0.393 0.404
0.370 0.380
0.915
0.826
0.952
0.668
0.605
3.846
0.877
2.853
2.867
2.914
3.854 TAU CHAN
3.894
1.908 TAU CHAN
1.815
ZRAT DHRAT
DELDH
ORDER COMMENTS
0.918
0.866
0.741
0.647
0.825
0.909
2.396 UNST ?
0.000 OSC.DIFS
3.722
3.608
3.959
1.314 TAU CHAN
1.353
DFLDH
ORDFP COMMENTS
0.92?
0.785
0.750
1.559 UNST ?
3.138
3.295
3.608
4.714
2.155 PR 33''
2.222
3.775 FIL.BYP.
3.8325
4.3?8
2.252
2.405
0.256
0.304
0.151
0.523
0.39
0.284
0.595
0.353
C.327
0.318
0.179
0.237
0.3f3
0.556
0.856
0.210
0.360
0.306
0.?70
0.?78
0.250
0.228
0.191
().128
0.716
0.597
0.373
0.332
0.758
0.313
0.297
C.268
0.205
0.635
0.550
0.401
C.372
0.665
ORAT DHRAT
0.349
0.360
0.327
0.442
0.343
0.520
0.691
0.301
0.316
0.270
0.433
0.292
0.552
0.794
0.756
1.402
0.779
0.776
7.711
RDER COMMENTS
0.93
0.309
0.119
0.781
7 PEAKS
T5AU CHAN
TAU=6.05
TAU=2.10
TAU=3.07
W' TAU?
W? TAU?
TAU=7.12
W INCFAS
I M t N T b
ALL PUINTS ARt TAOULATLU 3LLOuWIN UkuLR TAio.N
u wQ
RUN-PT
13
94
95
16
86
17
88
19
RUN-PT
OW
(W)
TIN
W
PO
(F) (LBM/HR) (PSI)
200
200
20C
200
200
200
200
200
61.2
61.2
61.7
61.8
62.0
62.0
62.1
62.0
QW TIN
(W)
(F)
E04
1 171
E04 92 171
E04 183 171
E04 94 171
E04
5 171
E04
6 171
E04 87 171
E04
8 I71
78.5
78.6
78.8
78.7
78.8
78.8
78.5
78.4
262.0
215.0
195.0
185.0
174.0
164.0
145.0
136.5
21.29
20.64
20.41
20.26
20.10
19.88
19.68
19.49
W
(LBM/HR)
Po
(PSI)
PI
DPlEX
(PSI) (PSI)
255.5
235.0
196.0
185.0
174.0
150.5
130.0
120.0
20.95
20.72
20.33
20.17
20.07
19.78
19.58
19.45
19.89
19.67
19.29
19.13
19.03
18.75
18.55
18.42
TES
RUN-PT
1
92
93
4
5
6
87
8
9
10
11
82
13
RUN-PT
1
92
93
4
5
6
P
L E N 6 T H b
T E S T
A L L
Pl DPIEX
XEX DTSUIN DTSUB8 Q/WHFG TAU
(PSI) (PSI)
(F)
{F)
(S)
20.22
19.58
19.35
19.21
19.05
18.83
18.63
18.45
TES
T
T
5.39
0.092 74.1
4.75
0.155 72.2
4.52
0.193
71.0
4.38
0.214
70.5
4.22
0.241
69.9
4.00 0.267
69.2
3.80
0.328 68.5
3.62
0.361 68.0
L ENG
T H
5.07
4.85
4.47
4.31
4.21
3.93
3.73
3.60
60.2
60.7
60.6
60.6
60.5
60.3
60.7
60.7
1
0.289
0.352
0.388
0.409
0.435
0.462
0.522
0.555
W I THoUT
0.115
0.138
0.193
0.212
0.235
0.293
0.360
0.401
55.8
55.1
53.8
53.4
53.0
52.2
51.9
51.6
43.3
43.3
43.4
43.4
43.4
43.4
43.9
43.9
0.253
0.276
0.330
0.350
0.372
0.430
0.498
0.540
L E NG
QW
(W)
TIN
W
(F) (LBM/HR)
PO
(PSI)
143
143
143
143
143
143
143
143
143
143
143
143
143
94.5
94.3
94.0
93.9
93.7
93.5
93.2
93.0
93.0
93.0
92.9
92.8
92.6
21.26
20.73
20.40
20.26
19.93
19.52
19.41
19.31
19.90
19.82
19.61
19.43
19.38
20.19
19.69
19.37
19.23
18.91
18.51
18.40
18.30
18.88
18.80
18.60
18.42
18.37
QW
(W)
T E S T
L E N
T H S
TIN
W
Po
P1 DPlEX
(F) (LBM/HR) (PSI) (PSI) (PSI)
114
114
114
114
114
114
109.4
109.2
109.3
109.1
108.1
108.8
420.0
364.0
?85.0
276.0
200.0
102.0
21.23
20.97
20.64
20.55
20.28
19.68
0.0C
3.50
3.50
2.40
2.52
2.40
7.65
7.65
WITHUU
TAU
(S)
Q/WFG
20.14
19.90
19.60
19.51
19.26
18.68
5.37
4.87
4.55
4.41
4.09
3.69
3.58
9.48
4.06
3.98
3.78
3.60
3.55
5.33
5.09
4.79
4.70
4.45
3.87
0.075
0.137
0.167
0.196
0.250
0.404
0.436
0.473
0.278
0.285
0.352
0.472
0.524
11 *
XEX
0.072
0.087
0.121
0.125
0.181
0.391
40.7
39.4
38.8
38.5
37.7
36.7
36.7
36.6
38.3
38.1
37.6
37.1
37.2
27.0
27.9
27.9
28.1
28.1
28.3
28.4
28.4
29.0
28.8
28.8
29.0
29.2
0.00
5.0C
2.65
2.68
2.70
2.62
6.00
6.10
2.75
2.74
2.68
6.06
6.05
0.158
0.221
0.252
0.281
0.335
0.490
0.523
0.561
0.366
0.373
0.440
0.561
0.e13
1.09
1.21
1.23
1.84
1.60
2.30
1.24
3.65
0.849
0.810
0.781
0.770
0.757
0.755
0.722
0.718
P Uw
XEXOTSUIN DTSU8B 0/WHFG TAU
(F)
(F)
(S)
TH S
1
AN
2
P1 DPlEX
XEX OTSUIN DTSU88
(PSI) (PSI)
(F)
(F)
343.0
245.5
215.0
193.0
161.5
110.5
103.5
96.5
148.0
145.0
123.0
)6.5
88.3
0.00
3.17
3.15
3.10
2.34
2.31
5.00
4.98
t U
H
WP/WA OPRAT
E
WP/WA DPRAT
1.11
1.17
1.23
1.38
2.35
2.50
1.61
4.40
0.874
0.812
0.782
0.780
0.769
0.754
0.736
0.731
0.397
?.10
Q/WHFG
25.6
25.1
24.7
24.1
24.4
22.0
0.103
0.119
0.151
0.156
0.216
0.423
0.00
3.5C
3.55
3.66
3.75
3.38
12.7
12.8
12.8
12.8
13.7
12.5
0.499
0.551
0.461
0.435
0.410
0.354
0.30n
0.377 0.286
1.05 0.844 0.698 0.605
1.14 0.796 0.589 0.448
1.14 0.797 0.550 0.392
0.788 0.524 0.355
3.10 0.788 0.484 0.297
3.20 0.769 0.419 0.?04
1.76 0.776 0.411 0.192
6.80 0.781 0.402 0.179
3.70 0.778 0.472 0.280
3.45 0.784 0.467 0.273
3.40 0.779 0.438 0.231
2.20 0.764 0.404 0.183
7.25 0.759 0.395 0.168
I T H OU
T
P
TAU
WP/WA DPRAT
IS)
W
0.638
0.598
0.523
0.502
0.480
0.434
T
P UW LK
WP/WA DPRAT 7RAT DHRAT
2
AH U
1TSUIN
DTSUB
(F)
(F)
3
0.582
0.812
0.810
0.798
0.804
0.798
5.20 0.805
1.03
1.06
1.14
1.80
3.10
0.889
0.695
0.825
0.853
0.836
0.685
0.679
0.795
0.843
0.845
0-,911.4
UW
L k
ZRAT DHRAT
0.661
0.631
0.584
0.578
0.542
0.475
0.440
0.386
0.301
0.291
0.227
0.105
DELDH ORUFR COMMENTS
0.801
0.810
0.803
0.778
0.751
1.589 TAU ?
1.851
1.837
2.321
4.422
Appendix
ENTHALPY TRAJCCTORY MODEL
**~**ut****
A COMPUTER PRCGRAM FOR THE
OF A BOILING CHANNEL.
INPUT
INFCRMATICN
FIRST
CARC
VERSION 8.2
********
IBM-360 TO INVESTIGATE THE
1/4/70
DROP
1
IMPLICIT COMPLEX (C)
REAL J, CEMM4), CJS, CPROJP, CYCLESCABS
MULF, MUGF
REAL MUL, MU.
INTEGER*Z NPLOT(60.3501
INTEGER*4 AERASE(105001
DIMENSION 8PL3T(120,'41,ZPLOT(121, XEXK(1201
EQUIVALENCE (NPLCT(1, N'ERASEIX)
CCMMON// D, OP, GL,ZHEZZMAX,
P1 , PRB, PEX, TIN. OTSUBB, DT
CCMMJN /STA20T/ OM(601, 8(6a), E(601, X(6019 10(601, PO(60,
DPAC(60), V(60), J(60)
I
DPDLFRIbOI, DPDLOR(60)
PDZDW, A02DW, SOOW
COMMON /OZDEOT/ PUHDw, ADHUW, SDHOW,
A
PDICWP. AUHUWP, SHOWP
1
,
PDP10Z, ADP1OZ, POPIDG, ADPIDG# PDPE. ADPE
2
CUHDWP, COPD12, COPICG, CDP1
COMMON/CCMPLX/ CDiCW,
GT40,1201, P(-4C9201
CCMMON /DYNPR/ 1(47,120),
COMMIN/PLOTOT/ XMAXJ(9), YMAXJ(9), XMINJI91, YMINJ(91, YRANJ(9)
STABILITY
ZPLOT(12) ; COORDINAft
ALONG THE CHANNEL IFT) AT WHICH THE TIME
VARIATIONS OF THE PRESSURE ARE TO BE ESTARLISHED; ANY
NUMBER OF VALUES UP TO 12.
AA I
VALUE OF T1E COEFFICIENT A ICHAPTERS). THE DEFAULT VALUE IS 0.8
LCORR
IIF'.EO.
1
USE LINEAR INTERPOLATION TO DETERMINE
PRESSURE AT EXIT; IF .EQ. S CALCULATE PRESSURE DROP IN LAST SEGMENT
SEPARATELY
NONLIN :
.EQ. 0 USE LINEAR ESTIMATE CF PRESSURE
IN SINGLEPHASE REGION FROM DZDWTF; IF .EQ.
CALL DYNPRI TO DETERMINE
PRESSURE DROP IN SINGLE-PHASE REGION.
WHEN THE SAME STEAUY-STATE INFORMATION (lW AND DPIEX) IS SUPPLIED
FOR TWO SUBSEQUENT POINTS, THE STEADY-STATE CALCULATIONS ARE BYPASSED FOR THE SECCND
IREPET -1 PERMITS TO REPEAT THE STEADY-STATE CALCULATIONS EVEN THOUGH
TAU DID 40T CHANGE
I0MIT -1 PERMITS TO BYPASS THE D2DWTF CALL
THE SAME PERIOD TAU
AND THE SAME STEADY-STATE INFORMATION WAS SUPPLIED BUT CALCULATICNS START AT A DIFFERENT PHASE, AS SPECIFIED BY TSKIP1
IF .EQ.
PRINTOUT OF THE PRESSURE, MASS FLUX, DZ/DT AND
COORDINATES OF THE MESH POINTS IS PRODUCED.
IPLCTI:
.EQ.
A PLOT OF THE PRESSURE AT THE BE IS PRODUCOD
IPLOT : IF ,EQ. 1 A MAP OF THE COORDINATES OF THE MESH POINTS
SIMILAR TO FIGURE 6.3 OFTHE TEXT IS PRODUCED
0-
F
0020W,
DZDT(40,120),
C
BGFIX*ALFAI * (-E.8628E-111*((1.D-XI(R.0-X)1(IS.0-ALFAI*ROL)
X*X/(ALFA*ROG))
DATA TWCPI/6.Z83185/
DATA WOLD, DPlOLO, TAUCLO /3*0.0/
O * 0.43/12.0
AREA - TWOPI * U * D / 8.0
ZLMAX - 13.0
KMAX = 120
K4AXL - KMAX - I
120
KMAX MUST BE .GE. 2 AND .Lt.
~
F * 25.0
F MUST BE SMALLER OR EQUAL THAN 25 LINES/FT FOR NPLOT(XX,350)
WHEN
IWRITE;
IF
1
1
SELONC CARC
TIN i INLET TEMPERATURE (F)
QWAY : GRCSS POWER (WATTS/TEST LENGTH)
PEX
EXIT PRESSURE IPSIAI
RELAMP I-RELATIVE AMPLITUDE OF THE FLOW OSCILLATION (UW/NOI
YSC : IF .GT. 0 DETERMINES THE SCALE OF THE PLOTSIMAX COORD.)
If .LE. ( SUPPRESSES THE PLOTS. IN THIS CASE ONLY THE MAXIMA
ARE DETERMINED BY PRTPL3
CYCLES : NUMBER OF CYCLES TO BE INVESTIGATED
TSKIP s DETERMINES THE STARTING POINT OF THE CALCULATIONS; PHASE
OF FLOW PERTURBATION IN CYCLES
CCMM(4) I CEMMENTS
C
C
C
1
2
3
s
THIRD AND FOLLOWING
MASS
4
5
6
7
LARUS
1
4SI
8
9
10
11
12
VALUE
13
14
15
16
LINES/FT
FORMATI12FS.2, F10.2, IX, 711)
FORMATITFX0.2. 2X. 2A4)
FORMAT(1' // 21X, 2A4, 5X, 2A4 /
F7.2, 5X, 'W =s,
TIN -,
21X,'INPUT INFORMATION
FT.1, 6X, DPIEX =', F7.2, 5X, *P1
2
FT.1, 5X, 'QWAV -',
FT.2, SX, /
3
4
T44, 'TAU =', F1.2, SX, *DT(SEC) -', F8.5, 3X, F5.2,
F5.2)
* CYCLES EXAMINED STARTING AT'
5
F8.3, 5X, *P88*
288 -,
FORMAT400SUBCSB RESULTS
F
F8.21
F8.3, 5X, *DTSUBB =
1
FORMAT('OSTAPR2 RESULIS : IMAX =', 15)
FORMAT('*O,15X,*CM, 1OX,'B*,I0X,'E*,10X,'X,10,X,*Z,1OX,'P'.
5X,'DPDLFR*,5X,'UPDL6R*,TX,'OPAC*,OX,V', 10X, 'J' /
1
2
(15, 5X, 3(IPEI1.31, OPF11.6v 2F11.3. 511PE11.3))l
T40
(MOD,ARG,DELI :',
FORMAT('OCZDWTF RESULTS :*, T20, 'OH
/
3F2C.5, * BTU/18M'
1
T20. 'DZ1, T40, 3F20.5, * FT' /
2
/I
T40, 3120.5.
3
T20, 'HP',
4
T20.
'M102',
T40, 2F20.5. 20X, ' PSI'
/
* PSI*
/
'PIDG,
T40, 2F20.5, 20X
5
T20,
',
T40, 2F20.5,20X,* PSI*,10X,'(A =,F5.2, *l' I
T20,
*PI
6
FORMATIOCDGPEAK =', F1C.1, * LSM/HR-FT2*, 5X, 'DZPEAK -* F8.5,
'
FG*, 5X, 'UUZDTP -'. FE0.5, ' FT/SEC' I
1
FORMAT(*0DYNPRI : Ptl,Kl 1* / 410F13.61)
FORMAT('1', 5X, 3014 /
FORMAT('ODYNPR2 : P1,K1
(PSIAI'
/
BX, 2016 / (15, 5X, 20F6.2))
1
/
*
FORMAT(*1DYNPR2 :Z{I,K
8X, 2016 / (15, 5X 20F6.311
I
FORMAT('0', 5X, 3014)
10**5(LRM/HR-FT2)' /
FCRMAT(IOOYNPR2 :G,K)
1X, 2016 / (15, 5X, 20(-5PF6.31))
1
IS ON A COLD SPOT
'I
FORMAT('OCZOWTF : BOILING SOUNDARY
/
FT/HRI'
FORMAT('ODYNPR2 : I2DT1,IK)
I
RX. 10112 / {15, 5X, 10F12.11I
1
RTU/LBM*
W I AVERAGE
FLOW RATE ILBM/HRI
OP1EX : PRESSURE DROP FROM STATION
TO EXIT (PSII
TAU : PERIOD OF THE CSCILLATION
TSRIPX 3 SAME AS ESKIP BUT FOR SUBSEQUENT POINTS. PERMITS TO BREAK
lHE ENTIRE CYCLE INTO MORE THAN ONE CALCULATIONS (SPECEFY
ADEQUATE TSKIP,
TSKIPI AND CYCLES)
PICOR : CORRECTION FACTOR TO MAXIMUM AMPLITUDE DF POSITION CF THE
BB PERTURBATION. DEFAULT VALUE IS 1.0
ADZCOR : CORREC TION (IN CYCLESI TO DELAY OF POSITION OF BB PERTURBATION. DEFAULT VALUE'IS 0.0 PEXCUR :
OF THt EXIT PRESSURE TO BE USED WHEN THE PRESSURE DROP
IN THE LAST StGMENT IS CALCULATED SEPARATELY. DEFAULT VALUE
IS EQUAL TO PEX. CAN RE USED TO SIMULATE THERMAL NON-EQUILIBRILUM
EFFECTS.
+
1
C
)
lFT
IT
18
FORMAT)'OREL. AMPLITUuE =',
F6.3)
FORMAT('0********-* FOR A RtLATIVE AMPLITUDE OF', F5.2,
9 THE RkLATIVE CENTLR OF VOLIUMt VtLOCITY AT THE BOILING',
/ 12X,'BOUNDARY MIGHT BECOME NEGATIVE', ..............
2
'CALCULATI:N DISCONTINUED')
3
' SEC')
FORMAT('OROILING TRANSIT TIME ='. F8.3,
FORMATI(IS, 3X, 6012))
FORMAT)'l' //
'OSAPE STEAnY STATE INFORMATION AND SAME DT USED NOW FOR
I
F6.2)
2
'A CIFFERENT PE-fOD OR TSKIP:'/'OTAU *,
2X, 2A4)
F-ORMAT(7F10.2,
2F11.3, 33X, 2(I1PE11I.1))FORMAT(15, 38X,
FORMAT)'0.........', 4A4, '.........'/)
FORMATf'OPktSSURE AT TMt EXIT FROM TIME', FT.4, * CYCLES TO TIME',
(5X, 15F8.4))
F7.4, ' CYCLtS'/
1.3 IN
,
INCREASED BY A FACTOR
FORMAT//'0******** OT(StC
'OGRDER TO REDUCE NUMBER OF MESH POINTS *********'//)
I
13)
*',
FORMAT(*OPEAK OF LXIT PRESSURE OUTSIDE RANGE, AT
FURMAT(*0*********** 10(', 12. ')SMALLER THAN 9.30207,
I
CALCULATICN DISCUNTINUED *=*********'//)
' , F6.3, 20X, F5.2, ' CYCLES EXAMI
' REL. AMPLITUDE
FORMAT)'0',
1NED NOW STARTING AT', F5.2)
FORMAT('CbXIT PRESSURE, PEX ' , F7.3, 10X, 'PRESSURE DROP IN TWO-'
, 'PHASE REGION, PBS-PEX -', F7.3 / 'OPEAK TO PEAK VARIATI',
1
PEX :', FD.4, SX, 'TWO-PHASE PRESSURF DROP tI,
'CNS :
2
F10.0, ' LBM/HR-FT2' 1
' PSI', SX, 'EXIT FLOW :',
F1C.4,
3
BOUNDARY CORRECTED',
FCMAT('O************ RDVEMENT OF
' EXTERNALLY :')
l
FORMAT(IX, 2A4, F6.1, F6.2, 2X, 6(F6.3, F5.3, 1X), lX, -5PF6.3,
SUMMARY
F6.3, 2X, F6.3, F5.3)
OPFS.3,
0 F
R E S U L T S , THRESHOLD POINTI',
FORMAT(/'OS U M M A R Y
2A4, ' AT', F7.2, ' WATT/TS'//
1
2X, 'POINT', T13, 'i', T18, 'TAU', T26, 'DMOW', T39, 'DZDW',
2
3
T52, 'DP1DZ', T64, 'DPlDG' , T76, 'DP1', T85, 'DP2(ENTH)',
T99, 'DG(FBK)', T110, 'RATID', T120, 'OPEX' /)
4
/ (5X, 20F6.3))
FORMAT('0EXIT QUALITY :
FORMAT('CEXIT PRESSURE DETERMINED RY LINEAR INTERPOLATION')
FORMATI'OEXIT PRESSURE DETERMINED BY CALCULATION OF PRESSURE DROP
LAST SEGMENT'/
F5.2,
2
' IN ALL CASES SEGMENT STARTED BEFORE ZCORR *',
ADJUSTED EXIT PRESSSURE, PEXCOR = ', F6.2, ' PSI
FT
3
4A')
1
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
C
C
C
DETERPINE PCSITIDN OF BOILING
OF
K
. .
.
. .
.
. .
. .
. .
W,
QWAV, DPIEX, PI, TAU, DTSEC, CYCLESTSKIP
WRITE(6,3) CUMM, TIN,
WRITE46,L7) RELAMP
DTSU8D
WRITE16,4) 18E, P8,
C
C
C
DETERMINE REFERENCE STEADY-STATE CONDITIONS IN THE CHANNEL . .
. .
CALL STAPR2(IMAX)
C
IF(IMAX .EQ. 1000) GO TO 114
IMAXI - IMAX - I
WRITE(6,5) IMAX
IM(II,t(I),E(i),X(I),Lt(I),PU(I)DPOLFR)I),
WRITE(6,6) (I,
I=1,1MAX1)
DPOLGR(I),DPAClI),V(I), J(I,
1
ZO(IMAX), PO(014AX), V(IMAX), J(IMAX)
WRITE(6,23) IMAX,X(IMAX),
C
C
C
CALCULATE THE TRAASIT TIME
IN THE BOILING REGION . .
. .
.
. .
. .
00 49 1 - lIMAX
IFIZO(I) .GE. 9.30207) GO TO 48
49
CONTINUE
WRITE(6,20)
GO TO 1111
48
BOLTRT = L(I-1)*DT+ALOG(I.+ODM(I-1)*E(l-1)*(9.30207-Z1-1))
/(L0(I)-L2(I-I))) / JM(I-1)) * '600.0
1
WRITE(6,19) BOLTAT
IF(IMAX-40) 112,112,115
114 DTSEC = CTSEC*1.3
TSEC - ISEC*1.3
115
CYCLES - KMAX*CTSEC/TAU
IF(IOMIT *LE. 0
1TSKIP - 0.5*(1.0-CYCLES) - 0.5*SDLTRT/TAU
WRITE16,26)
GO TO 113
IMAX
1X,
IIN
C
.
UTSEC - TAU 0 CYCLtS / FLC4TIKMAX)
OT - OTSEC/1600.0
113
C
80LING
1
.
C
C
F11.4,
1
BDUNDARY
CALL SUaCEB
C
C
CALL ERASEIZ,4800, D0OT,4800, G,4800, P,4800)
112
C
C
C
REAC AND PROCESS INPUT INFORMATION
ZCCRR
.
.
. .
. .
. .
.
. .
. .
READ(5,2)
1110 READ)5,1,END-9999) ZPLOT, AA.
LCORR, NONLIN, 10DIT,IREPET, IWBITE,
1
IPLCT1,
2
READ)52,END*9999) TIN,QWAVPXRELAMP,YSC,CYCLESTSRIP,
1
CCPM(1),COMp(2)
IF(TIN .LE. 0.0) STOP
A - 0.8
IFAA .NE. 0.0) A - AA
WRITEI8,33) COAMI), COM12), QWAV
ADZCOR
POZCO,
0PIX, TAU, TSKIPI,
1111 RCAO5,22,ENDO9999F
PEXCOR, COMM(3), COMMI4)
I
.LE. 0.0) PEXCOR * PEX
IF(TAU .LE. 0.0) GO TO 1110
IF(IOMIT .LE. 0) GO TO 117
TSKIP - TSKIPI
I1MW .NE. WOLD) .0R. IOPlEX .NE. DPIOLD) .OR. IIREPET .EQ. 1))
117
GO TO 111'
I
WRITE(6,21) TAU
WRITE16,29) RELAMP, CYCLES, TSKIP
GO TO 112
IPLOT
W,
IF(PEXCOR
.
IF((ICMIT.GI.)0.AND.(TAU.EU.TAUOLO).AND.(W.EQ.WOLD).AND.(DP1EX.EQ.
60 TO 116
OPIOLD))
I
C
C
C
C
CALCULATE PEATUReATION tF THE POSITION OF THE RB AND SINGLE-PHASE
REGION DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . .
CALL
DEbTFTAUICOLD)
C
IF(ICCLU .EQ. 1) WRITE(6,15)
PDz1W, AOZuW, SDZDW,
WRITE46,7) POMOW, ADHDW, SDHDW,
PDFCWP, ADHUWP, SOHOWP, POP1DZ, ADP1UZ, POPIG, AOPlOG,
POPE, ACP1, A
2
1
C
C
C
C
C
-
CORRECT IF NECESSARY THE AMPLITUDE OR THE PHASE OF THE POSITION OF OR
PERTURBATIOn . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
IF(POZCCR .EQ. 0.0) PCZCOR = 1.0
IF(IPDCOR .EQ. 1.0) .AND.IADZCOR .EQ. 0.0)) GO TO
ROL = TWCPI*ACLCOR
CDZCOR v CMPLX(COS(ND),SIN(RUZIb*PPLZCUR
CDZOW = CCZCW*CCZCOR
COPIOZ = CCPIDZ*COlCOR
CDP& - CCPI0Z * COPIDG
118
C
C
111
P
- PEX + DPlEX
QP = (3.412/1.291667)OQWAV
QTP - QP/AREA
G0 = W/AREA
CAOSCDZDW)
AMO0(1.0-ATAN2AIMAG(CDZDW),REALCDUZDW))/TWCPI),1.O)
TAU * ADZDW
PDPIDZ - CABS(CDPID)
ADP1OZ = AMUDI(1.O-ATAN2)AIMAG(CDP1DZ),REAL(CDPI0Z))/TWOPI),1.0)
PDLDW
ADZOW
SOZDw
=
=
=
(CYCLES)
)I
- CAS(CDP1
PQPl
))/TWOPI),1.0)
I,REAL(COPI
* AMO((l1.0-ATAN2(AIlAG(COP
ADP
WRITE(6,31)
PUIOW, ADZDW, SOZUW,
WRITE(6,1I PUHDW, ADHa, SDHDW,
PDIDWP, ADHDWP, SHUdP, POP1DZ, ADP10Z, POPIG, ADPE0G,
S
POP1, ADP1, A
2
C
C
C
C
I(PSI
(CYCLES)
WRITE(6,14) (Li L -1,K20,KDK),
It, IG(II,K), K-,K20,KDKI, 1.1,IMAX)
-WRITE(6,16 (L, L =1,920,KDKI,
K-1,K10,KDK2I,
(I, (ZDT(IK),
1
WRITE16,24) COMM
1
I-1,IMAXI
C
187
ZI1,K
GENFRATE C(1,K),
AND ODZOT(I1,K) . . . .
. . . . . . . .
. .
C
118
OGPEAK - RELAmP*GC
DZPEAK = RELAMP* PDZOW
DOZOTP IFI(GO-OCPEAK) * V(I),DUDTP) 50,50,51
WRITEI6,10) RELAMP
GO TO 1til
T - TWCPI * ADZDW
DTOIN = CTSEC/TAU
UTIME * 7tAPI*DTUIM
TINE = -OTIME + tSKIPTTWOPI
0ZPEAK*T4OPI*3600.0/TAU
50
51
C
99
C
-
DO 199 K
'IFIZMAX-Z(IMAX,K)
ZMAX - J(IPAX,#K
CONTINUE
291
299
DO 299 K -1,KMAX1
IFIZlEE) - ZMIN)
ZMIN - Ztl,K)
CONTINUE
C
C
C
PRODUCE A MAP OF
C
352
389
IF(NONLINI 60,60,61
C
C
63
C
61
C
62
C
C
C
TPI * TWCPI * AGP1
TIME = - CTIME + TSKIP * TWOPI
1,KMAXI
00 43 K
TINE * TIME + CTIME
P(1,K) - PB + RELAMP*PJPI*COSITIME-TP1)
GO TO 62
OYNPRIIKMAX)
IF(IWRITE .GT. 0) WRITE(6,9I
1,KMAX1
109
(P(I,K),
K
=
1,KMAX
C
C
C
C
1I*OTDIM
410
KMAXI,2,
C
KMAX3 a
EPAX2
-
C
IFI(WRITE
,LE.
KMAXI,0,
KMAX,4,
YSCII
. .
471
479
I
1,IMAX
00 399
*
KKK - KMAX - I
00 399 K -1,KKK,KDK
NPLOT(K,Z4I,Kl*F+1I)
=
I
+ 10
TO 187
481
489
KOK
C
I=leIMAX)
I-I, IMAXI
IMIMIN - ZMINMF - 1
IMAMAX = ZMAX*F +
IFIZMAX .GT. 13.96) IMAMAX = IFIX(13.96*F)
WRITE(6,104 (I, I =1,KMAX,KD3K2l
1
WRITE(6,20) tI, (NPLOT(K,II, K -1,601,
WRITE(6,13) (I, I =1,KMAX,KDK2)
WRITE(6,241 CCMM
1
ZMIN * 1.CE+50
D0 479 K =1,KMAX2
IFtZIIMAXKI-ZMIN) 47T1,471,479
ZMIN * ZIIMAX,K)
CONTINUE
IF(ZPLOTtll .LE.
1
* KPAX1/20 + 1
KDK2 - 2*KDK
K20 * 19*KOK * 1
10 - 9*KCKZ + 1
IL, L -1,K20,KDK),
WRITEI6,12
K-1,K20,KDKI,
(I,
(ZII,K),
1
WRITEt6,111 IL, L =1,K20,KDK)i,
(I, (PII,9), K-l,K20,KDK),
1
389 1 = 1,IMAX
KKK - KMAX - I
00 389 K - 1,KKkKOK
IFIZlIK) .GE. 13.961 GO TO 389
NPLCT(KZ(IK)*F1I) - I + 10
CONTINUE
GO TO 400
D0
C
C
C
C
KMAXKMAX2)
01 GO
THE MESH POINTS . .
=
IMIMIN,IMAPAX)
FIND TIME VARIATIUNS OF PRESSURE AT GIVEN LOCATIONS ZPLOT(Il
FIND MIN VALUE OF ZIIMAX,Kl
+ TSKIP
USE THE ENTHALPY TRAJECTORY MODEL FOR THE DYNAMICS OF THE TWC-PHASE
*
.
REGION . . , . . . . . . . . . . . . . . . . . . . . . . . .
CALL GYNPR2(IMAX,
C
399
40C
YSCI - -1.0
00 109 K *
BPLOT(K,1) - (K RPLDTIK,2l * P(I,K)
CALL PRTPL3I1,RPLOT,
THE COORDINATES Of
C
PLOT THE PRESSURE AT THE ROILING BOUNDARY VERSUS TIME.
C
C
C
C
C
351
CALL
YSC1 = 1.0
IFIIPLOTI .LE. 0)
291,291,299
KOK
kNAX1/60 + 1
2*KDK
KDK2
IF(ZMAX - 13.96) 351,352,352
IF A NONLINEAR TREATMENT OF THE SINGLE-PHASE REGION IS SPECIFIEW'
DYNPRI IS CALLEO, OTHERWISE THE DZDVTF RESULTS ARE USED
TO CALCULATE THE PRESSURE DROP IN THE SINGLE-PHASE REGION
60
191,191,199
. .
.
CALL ERASEINERASE,105001
ERASE IS A CENTER SUPPLIED SUBROUTINE FOR ZEROING LENGTH *4 ARRAYS
C
C
T)
OUZOTP = DDZDTP/3600.0
WRITE(6,81 DGPEAK, CZPEAK, ODDOTP
C
C
C
C
C
C
=I,KMAX2
191
199
C
DU'9
K *l,,jMAX
TIME
TEME + OTIME
GO + DGPEAK*CCS(TIME)
G(1,Kl
141,K) - Z88 + DLPEAK*COS(TIME
-DDZTP*SIN(TI4E-T)
D20T44,KI
GG TO 410
IMIPLOT .LE. 0)
ZMAX = -1.CE50
ZMIN = 1.0E50
C.0) GO TO 500
FINC MAX VALUE OF Z(1,K1
ZPAX - -E.OE+5,0
00 489 K -1,KMAXZ
IFIZMAX - AI1,RI1481,489,489
ZMAX k ZI,K
CONTINUE
00 499 L
ZP - ZPLCTILI
0,0) GC TC 500
IF(IP .LE.
IF(ZP *GT. 2MIN) GO TO 499
IF(ZP .LE.
ZMAX) GO TO 499
=1,12
M =0
IAX
496
C
C
C
BPLOT(1M,21PII-1,K+I)+(P(I,K)-P( I-1,K+1))*(ZP-Z(I-1,K+1))/(Z(1,K)
497
494
498
499
500
C
-
Z(I-1,K+1)F
I
BPLOT(M,1) - (K +
- 21 * uTDIm + TSKIP
GO TO 498
CONTINUE
M- M - 1
CONTINUE
CALL PRTPL3(IFIX(100.*ZP+.5),BPLOT,
M,2,
CONTINUE
CONTINUE
WOLD
M,0,
KMAX,4,
C
C
C
=w
FIND FLEW AND PRESSURE VARIATIONS
AT EXIT AND UP
IN TWO-PHASE REGION
ZP - 9.30207
PRBEX- PBB-PEX
C
C
C
C
ADD ENTIALPY VARIATIGNS AT ACTUAL POSITION OF R3 TO ENTHALPY AT
MESH POINT If
KMAX2
M + 1MAX - II - I
DU 559 K - 1,M
GIl - G(I,K)
D1 - ZP - Z(II#K)
TIME = TWOPI*TSKIP + (K-1)*JTIME
+ RELAMP*PDHDWP*COS(TIME-ADHDWP*
XEX a Xi + IQTP*I(0-0.26)/GII
TWOPI)P/HFGEX'
1
XEXKIK) - XEX
XAV - 0.50(Xl1 + XEX)
XTTtX = XTTi*tEt.0-XEXI/XEXI**0.9
XTTAV * C.5*IXTTII + XTTEX)
ALFAAV * ALFAF(XTTAV)
ALFAEX = ALFAFIXTTEX I
XUMUL- (I.0-XAV)*D/MUL
OPFRIC - FI2LFIXTTAV)*(-3.3256E-11)*IDE/DI*FF(GII*XDMUL)*GI **2
*tl.0-XAV)**2 / ROL
I
DPGRAV = -CL*IROGROL * ALFAAV + ROL3/144.0
DPACCE = (OGFIXEX, ALFAEX) - RGII)*GII*GII
dPLOTIK,1) - (K+I-21*DTDIM + TSKIP
8PLCT(K,2) - fP(II,K)+ OPFRIC + DPGRAV + DPACCEI/PEX
BPLOT(K,31 = GII/GO
9PLOT(K.4) - PiIK11-1 - BPLOTIK,2)*PEX)/PR8EX
559
C,
IF(LCORR .EQ. O1 GO TO 550
IF(ZP .GT. ZMIN) GO TO 550
C
C
C
546
547
544
548
C
USE LINEAR
INTERPOLATION TO
DETERMINE EXIT PRESSURE.
WRITEI6,351
M 0
DO 548 K = I,KPAX3
M =M
+
DO 547 1 - 1,1MAX
IFIZ(I,K) - ZP) 547,546,546
IF (Z(I-I,K+1I .GT. ZPI GO TO 544
CPROP = IZP-Z(I-iK+1))/ (Z(IK)-Z(I-I,K+1I))
BPLOTIM,I) = (K + 1 -2) *DTUIM + TSKIP
BPLOT(Mt2J - IP(I-i,K+i) + CPRUP * (P(I,KI-P4I-1,K+1)I)/PEX
BPLOT(M,3) = (G(I-1,K+1I + CPROP*(G(I,K) - G(I-1,K+1)))/GO
IP(I1,K+1-1)-BPLOT(M,21*PEX)/IPBEX)
BPLOT(M,41
GO TO 548
CONTINUE
M = P - 1
CONTINUF
CALL PRTPL3(93C,RPLOT,
,4,
M,0, KMAX,4,
YSCI
WRITE16,24) CCMM
WRITE46,25) BPLCT(I,1), BPLOTIM,I), (BPLOTIK,2), K =1,M)
PEXRAN = VRANJIII *PEX
GEXRAN = YPANJ121 * GU
DP2RAN = YRAAJ(3) * PROEX
WRITE6,301 PEX, PBSEX, PEXRAN, UP2RAN, GEXRAN
C
M,4,
CALL PRTPL3(930,BPLOT,
WRITE(6,24) CCMM
PEXRAA = YRANJ(I) *PEX
GEXRAN = YRANJI2) * G3
* POBEX
DP2RAN =
WRITEIE,3C) PEX, PBREX, PEXRAN, DP2RAN,
WRITE16,34)(XEXK(K),K - 1,M)
YPAAJ(3)
600
550
C
C
C
C
C
IFILCORR
.GT. 0) GO TO 600
EVALUATE PRESSURE DROP IN LAST SEGMENT INDEPENDENTLY
UNTIL ZMAX .LE. CHANNEL
FUR IM4AX, IMAX-1, ...
FIND
LENGTH.
ZMIN, ZMAX,
551
552
II = IMAX + 1
11 = I1- 1
ZMAX = 0.0
00 552 K = 1,KMAX2
IFtZfII,K) .GT. ZMAX) 14AX - Z111,K)
CONTINUE
IF((ZMAX.GT.ZCORR).AND.(II.GT.1)) GO TO 551L
IFIZMAX .Gt. ZP) GO TC 600
M,0, KMAX,4,
YSC)
GEXRAN
C
C
C
ESTIMATE PROPERTIES AT AVERAGE PRESSSURE
+ PtXCOR)
PAV - 0.5*IPO(II)
MUL * MULF(PAV)
MUG - PUGF(PAV)
ROL = RCLFIPAVI
ROG - ROGF(PAVI
ROGROL * ROG - ROL
XTTI - (ROG/R(TL)**0.5 - (MUL/MUG)**O.l
XTTII * XIII *tE,0-XIII/XIII**0.9
ALFAII - ALFAFtXTTII)
BGII - 8GF(XIR, ALFAIII
YSC)
UPIOLD - OP1EX
TAUOLD - TAU
C
C
C
WRITE(6,361 ZCCRR, PtXCOR
HFGEX = bfGF(PEXCOR)
HEX - HLFIPEXCOR)
XII = X11)
HII = HLF(POIIIII + XII *HFGFIPO(II)
X1 = EHII - FEX)/HFGEX
AT ZP BY LINEAR INTERPOLATION
DETERMINEPRISSUE
I
DETERMINE EXIT WUALITY ACCURDING TO PEXCOR, CORRECTED EXIt
PRESSURE 'TO TAKtt INTO ACCOUNT Sflt THERMAL NON-EQUILIRRIUM
C
C
C
00 498 K - i,KMAX3
M
M + I
DO 49T 1 - 1,
IF(ILK) - ZP) 497,496,496
IF#Zil-1,K+11 .GT. ZP) GU TC 494
C
C
C
C
CONTINUE
CALCULATE OPEN LOCP
TRANSFER FUNCTION
AVP2E * XPAXJIJl
PDP2E - 0.5*P?RAN / RELAMP
C
ROP2 = TWOPI*ADP2E
COP2E = CPPLX(COS(-RDP2ISIN(-ROP2I)*PDP2E
CDPIt -- CCP2E
COPIG - CDP1E - CUPlCZ
/ CUPOG
COGFDR
PDGFEB = CABSICDGFU8)
A3OD((I.0-ATAN2(AIMAGICL)GFDB),REAL(CDGF09)/ITWOPI),I.0)
ADOGFDb
RATIC = PCGFDR / GO
WRITE(8,321 CCMM(3), C0M(4), W, TAU, POHLIW, 4M4OW, PDZDW, A0ZOW,
PDPIDZ, APICZE, PJPiG, AOPI0G, PDPI, ADPI, PDP2E, ADP2E,
SUMMARY
PDCFC , ADGFOB, RATIO, PEXRAN, XMAXJ(I)
2
GO TO 1111
9999 STOP
END
=-CDPiG*GO
1
P(1+1) = PI1) + DPFR + JPGR
PSATII+I) = PSATFit(Il+1))
SUBROUTINE SUBCBB
Z(1+1)
VERSION 3.1,
C
C
C
C
C
C
C
C
C
11/3/69
TAKES INTO ACCOUNT C-LD SPOTS
CALCULATE SURCOOLING WITH RESPECT TO BOILING BOUNDARY FDR EXPERIMENTAL
INLET.
PClNTSiSTARTING PRESSURE DROP CALCULATIONS
INPUT I CP IBTU/HR-FTh# G (LBM/HR-FT2), PIN - Pi (PSIA), TIN (F)
PB6 IPSIA), DTSUBB IF)
OUTPUT : 288 (FT),
HLTF, PSATF, TF, FF KLF
RULTF,
FUNCTICNS REUIRED
FROM
9
MULTF,
I
REAL PULB6MULw,MULtF, KLF
REAL QDISTRIP)/?*T.0/
T1161, P(L6)o PSAT(16), H(16)
21(15),
DIMENSIEN DISTR(15,
COPRON // Do 6P, G, Z8, LMAX, PIN, P88o PEX, TIN, DTSUBB, DT
DATA DLH+ DLU /1.125, 61666667/
C
C
C
*********
FRECN -
113 PROPERTY FUNCTIONS
******
ROLTFI/1
PSATF
TF
RLF(T)
KLF
*OS165A*O*D*G
IENERATE CORRECTED FEAT
.
FLUX DISTRIBUTION..
. .
.
.
..
RATIO * IU00 + DLH)/ULH
2,14,2
DC 98 J
GQISTRJ/2) * RATIO
DISTR(J)
98
C
C
C
GIVE INITIAL VALUES At HEATED SECTION
INLET .
.
..
.
.. ..
+
DL
BOILING BCUNDARY BY ITERATION .
DLP
- OL
DLP - DLP * IPWII - PSATII)) /
* OLP/DL + H(I)
H(I1+1) = DH
TF(HIl+1))
it1)
TAV - 0.5 * (TI)) + t1+1))
ROL - RCLTF(TAVI
PULB
=
. .. .
. .
(P(I)-P(I+I1)+PSA(1+1)-PSAT(I))
PLLTF(TAV)
TWAV - TAV + DTFILM
MULTF
HLTF 1/1
C
W
211)
FIND EXACT LCCATICN OF
2
6
1.0 / It + 459&61
TIT)
.D305453 E 03 - .7105255 E-01*T - .6448694E-04*t*T
ROLTFITI 4
+ .5585921 E 04*TI(T)
PULTF(T}* EXP( - .3483695 E 01
+ .3577018 E 09*TI(TI**3)
.1950534 E CT*TI(T)**2
I
HItFiT)- *.119482E GE + .1961794E 00*T 4 .126I04tF-03*T*T
I.0**133.C655-4330.98/(T+459.6) - 9.2635*ALCGIO(t+
PSAtF(T)
+ 0.0020539 * (T+459.6)1
459.6
1
- .1141203E-O*H*H
- .4C8?034E 02 4 .51866R1E 01*H
TFIH)
* C.0475 - 6.?666TE-05* T
C
C
C
=
IfT-(Pl+1) .LE. PS&tIl+1)) Go TO 2
IFIl *Lt.15) GO TO I
DTSU8 = t(l6) - TIN
P88 * P116)
288 - 9.C41668
WRITF(6,9)
: BOILING aCUNDARY IS OUTSIDE HEATEDN
SUbC B
FORMAt(00********
//)
t* SECTION'
I
RETURN
1/1
1/1
= PULTF(TWAVI
PULW
REL
G * 0 / PUL3
6
DPFR w (-3.3256E-11)*(DLP/)I*FF(tL)*G*G*IIMULW/UL8I**0.14)/ROL
OPGR = -OLP *ROL /144.0
DPI- +-DPGR
a PI T)
P(1+1
PSAt(I+
- PSATFIT(1+1))
IFIABS(P(I+1)-PSAT(I+1)) .GT. 0.01) GO TO 6
DTSUBB -k 1T+1 - TIN
P141)
P88
+ DLP
Z88 = 2)
RETURN
END
.
SUBRCUTINE CZOWTF(TAU, ICOL)
VERSION 3.2,
til) = 0.0
Hit) = HLTF(TIN)
- TINR
TI()
Pit) -PIN
PSAT11) - PSATFWIN)
C
C
C
CALCULATE BULK TEMP. AND SATURATION TEMP. AT THE END Of EACH INTERVAL
1
3
I
1
0 +
=1.+1
If-IMCCII,)I
UL = DLH
OH
H(I+1)
3,304
CL*QP*DISTR (I) / W
i-I) + OH
*
Tili+1) = YTPINI+1-)
TAV = 0.5 * (TII) +
ROL = ROCLTFITAV)
4
OTFILM
5
T(I+1))
- MULTFITAV)
MUL
OTFILM = 1282.*D.L)*QP*OISTR(I)*MULB**0.4/(G**0.8*KLF(TAV)**0.6)
tAAV- TAV + OtFILM
MULW * AULTFITWAV)
GO
5
0L - DLU1
IA UL * 0.5*DLU
IFI .EO.
DH a 0.0
H(11)
& H()
TAV
TIE)
T
uil1TAV
ROL * RTLTF(TAV)
AULB- PULTFIAV)
* 0.0
MULW
VLLS
P
MULBREL =G * D
OPFR = (=3.3256E-1t)*(CL/0)*PF(REL)*G*G*(IMULW/MULBI*=0.141/ROL
nOGR = - T" * ROL / 144.0
TC
12/0/69
SUBROUTINE OZOWTF CALCULATES THE AMPLITUDE AND PHASE OF THE
BOILING BCUNCARY USCILLATION USING A LINEAR MODEL.
IN THIS VERSICN THE CCLO SPnTS IN THE WALLANU THE HEAT FLUX DISTRIBUTICN ARE TAKEN INTO ACCOUNT
THE PRESSURE CROP VARIATIONS IN THE SINGLE-PHASE REGION ARE ALSO
CALCULATED.
INLET VARIABLES
TIN t INLET TEMPERATURE IF)
DTSus84 SUBECCULING WITH RESPECT TO THE BOILING BOUNDARYSAT.
PL: PRESSLRE AT INLET CF HEATED LENGTH (PSA)
QP 1 AVERAGE LINEAR HEAT RATE (BTU/HR-FT)
AVERAGE MASS FLUX (LBM/HR-FT2I
G
DISTANCE OF BOILING BOUNDARY FROM INLET (FT)
2881
TAU
PERICO Of SINUSOIDAL OSCILLATION (SEC)
TEMP IF)
t
QDISTRIt) DIMENSIONLESS POWtR
A : COEFFICItNT DEFINED BY 0Q/Q
DISTRIBUTION (INLET
= A * OW/W
TO EXIT)
OUTPUT t
DHDW 1 INLET FLOW TO ENTHALPY AT STEADY STATE LOCATION OF 88
EFFECT), IBTU/LBM)
(UITH
ZOW : INLET FLOW TO POSITION OF ROILING BOUNDARY (ALL EFFECTS
(FT)
INCLUDEC)
FLCW TO ENTHALPY AT MOVING POSITION Of RE, (STU/LBM)
DHDWP :
: MOVEMtNT OF THE B4 TO PRESSURE DROP IN THE SINGLE-PHASE
REGION (PSI), FRICTICN AND GRAVITY EFFECTS.
OP1OG : INLET FLOW TO PRESSURE DROP IN THE SINGLE-PHASE REGICN
(PSI), FRICTION ANC INERTIA EFFECTS
OP
: INLET FLCW TO TOTAL PRESSURE DROP IN THE SINGLE-PHASE REGION IPSI)
(F THE BB IS IN THE HEATED ScGMENT, .t . I IF THE 88
:
EQ.
IS IN THE COLD WALL
WALL
DPIUZ
ICOLD
INLET
0
C
C
C
C
C
C
THE
THE
THE
THE
PREFIX
PREFIX
PREFIX
PREFIX
C
P
A
S
DENOTES
GENCTES
DENVTES
DENOTES
THE
THE
THE
THE
COMPLEX JUANTITY
MAGNITUDE (PEAK VALUE)
PHASE (IN CYCLES)
DELAY (SEC)
881
NTS GU TO
882
111+2) - L(1+1) + DZU
IF(ZAIB-Z(1+2)) 488,q83,88
NTS - I + 1
GC TC 89
CONTINUE
NTS = NTS + I
Z(NTSXI - 2P3
[COLO = MCD(NTS.2)
00 87 I = 2,NTS1
COEXP(I) - CEXP(CK*ZIll)
883
IMPLICIT CCPPLtX (C)
REAL PULTF, KW* CAdS, KLF, MULB
DIMENSICA
COEXP(16), DISTR(15), Z(16)
COMMON// 0,
.JPG,
LMAX, Pl, P88, PEX, TIN, DTSUBB, OT
COMMON /CLOWOT/ POHDW, ADHDW, SDHW,
PDZOW, ADZDW. SDZDW.
I
PDHDWP, AOHDWP, $DHDWP, A
2
, POP10Z, AUPECZ, PDP1UG, ADPIG, PDPE, AOP1
COMMOA /CCMPLX/ CUHCW, CDZ.., CDHOWP, CDPEOZ
CDPE0G, COPE
DATA DISTR/15*0.0/, WUSTR /7*1.0/
DATA PITWOPI/3.141593, 6.283185/, CI/(0.0, 1.0)/,
ALPHA, KW, THICK
/0.022, 0.75, C.004/
I
2
, DlH, OL/1.125, .1666667/
88
81
QDISTR(7),
LU6,
C
C
C
PROPERTY FUNCTICNS
RE;UIRE:
ROLTF,
MULT,
DHLF,
KLF,
TI(T) - 1.0 / (IT * 459.6)
MULTF(T)= EXP( - .5483695 E 01 + .5585921 E 04*r[(T)
- .1950534 E 07*TI(T)**2
+ .3577018 E 09*TI(T)**3)
ROLTF(T) .1035453 E 03 - .7105255 E-01*T - .6448694E-04*T*T
HLTF(T) = .81194F2E 01 + .1961794E 00*T
+ .1261047F-03*T*T
KLF(Tl = 0.0475 - 6.76667E-05* T
AVERAGE TEMPERATURE
AND OENSITY
M.I.T.
MULTF1/2
MULTF2/2
ROLTFE/E
HLTF 1/1
KLF 1/1
-RCLAV
811
C
C
C
PBBCAL
DHLBB
*
(PSI)
Pl + DPER + CPGR
DhLFAPS8CAL)
C
W = C.7e54 * 0 * C * G
CCMEGA = 3600.0*TWOPI*CI/TAU
SPEHEA = IHLTF(TAV + 5.01-HLTF(TAV - 5.0)) * 0.1
HFC = 0.C279*W**O.8*KLF(TAV)**0.6*SPEHEA**0.4 /
E
(0**1.d*MULB**O.4)
C
IE/HRI
STL/LB--
GP
ABC / h
CI - CSQRT(CUMEGA/ALPHAl
C2 = Cl * THICK * CI
C3 = CI * CCOS(C2)/ (KW * CSIN(C2 * CE)
C4 * Pt * D /(1t.0/HFC+C31wW*SPEHEA)
C5 - RCLAV*CCMEGA/G
CK - CS + C4
C6 - A * C3 /(1.0/HFC + C3)
CT = (1,0-Al * ABC
CB = C& 8 ABC
CL - C7 + C8
CGI = CL/CK
CG2 = -CSLM/COLXPANTSI)
CCHDW = CGI * CG2
(STU/LBI
CALCULATE THE
INLFT
FLCW TO POSITION OF THE
8.8. TRANSFER FUNCTION
1.8*(DPFR-1.735E-04*MULTF(TIN)**0.2*W**.8/ROLIN4*DHLBR
(RTL/LB)
(BTU/LB)
(STU/LBO
(BTU/LB)
(IFT)
PDZDW = CABSICCLDwI
ADZDW = -ATAN2(AIMAG(CDZOW),REAL(CDZDWIl/TWOPI
IF(AUZ'DW .LT. 0.0) ADD = 1.0 + ADZDW
SDLON - TAU * ADZDW
C
C
C
C
CALCULATE TFE INLET FLOW Ti ENTHALPY AT THE ACTUAL POSITION OF
THE 8. 8. TRANSFER
. . . . . . . . . . . . . . . . . .
FUNCTION
COHDWP = CDFIW + 81 * CDZDW / ZBB
PDHDWP - CABS(CDHDWPl
ADHDWP = -ATAN2(AIMAG(CDHOWPI,REAL(CDHDWP))/TWOPI
IF(ADHEDWP .LT. O.C) AO14OWP = 1.0 + ADHOWP
SDHDWP = TAU * ADHDWP
C
C
C
CALCULATE TE PRESSURE DROP T.FS.
.
(8TU/L8)
IN THE SINGLE-PHASE
REGION.
CDPIDI - (-DPFR - DP(.R)*CUZDW / Z88
COPEFR = -CFR/DHLBB
CJP1IN - -CIN/DHLeS
CCPL0G = COPIFR + CDPEIN
CDPE - CCFICZ + CCPEIC
(PSI)
POPI
PDP1FR
PDPIIN
POP10G
POPE
AOP10Z
ACPlFR
ADPAIN
ADPIDG
ADP1
(PSI)
(PSI)
(PSI)
(PSI)
(PSI)
(CYCLES)
(CYCLES)
(CYCLES)
(CYCLES)
(CYCLES)
(PSI)
(PSI)
(PSI)
(PSI)
C
FEATtD
(CZU + DZH)I/CLH
RATIO
ZlE) = 0.0
Z(21 = 0.5*DZU
2,14,2
D 88 1
DISTR(I) = QDISTR(I/2) * RATIO
Zf1+1) - Z(I) + DZH
IF(ZB-L(I+13) 881,8819882
CJEXP(II)
C
FIND IN WHICH TEST SECTION THE BOILING BOUNDARY IS LOCATED
GENERATE
AND UNHEATED SECTION LIMIT CDORDINATES AND CORRECTED
FLUX DISTRIBUTION
FEAT
-
FLCW T9 ENTHALPY TRANSFER FUNCTION...
CFR=
CIN= -TWCPI*G*(Ze+2.3)*DHLBR*CI/(TAU*3600.0032.17*144.0
BI - 288 * QP *
DISTR(NTS) / W
82 = -DhL8*(DPFR+DPG4I
CDZDW - (-COHOW + CFR+ CIN)*ZRB / (81 + 82)
PULB)**0.14/ROLAV
1
CALCULATE THE INLET
PDHDW = CABS(CDHDW)
ADHOW = -AT8N-2IAIMAG(CCHDW),REAL(COHDWI)/TWOPI
IF(ADHDW .LT. 0.01 ADHOw = 1.0 + ADHDW
SDHDW = TAU * AHW)W
IN NON BOILING REGION
*
/ 144.0
DTFILM = (282.0*1.2916671*5P*SMULB**0.4/(G**O.8*KLF(TAV)**0.6
UPFR- (-3.3256E-11 *ZB8/O*FF(G*D/MUL8)*(,*G*
(MULTF(TAV+DTFILMI/
EVALUATE INTEGRAL
CSU4 =0.0
DO 86
= 2.ATS,2
CSUM * CSUM + DISTR(I)*(CEXP(I+1)
C
CALCULATE CRAVITY AND FRICTIONAL PRESSURE DROPS IN NON BOILING,
HEATED REGICN
. . . . . . . . . . . . . . . . . . . . . . . . .
DPGR =
89
I
86
C
C
C
TAV * TIA + 0.5*OTSUfd
ROLAV = ROLTFITAV)
ROLIN = RCLTFITIN)
MULB = sULTFITAV)
C
C
C
C
87
C
C
C
M.I.T.
HLTF
1
C
C
C
*******
I
C
RE TURN
END
- CABS(CUP0LI
= CABS(CDPlFRl
CABS(CJPIIN)
- CABS(CDPI0G)
I
- CARSICDP
- AMOU(I(.0-ATAN2(AIMAG(CDPIOZIRAL(CDP10ZI)/TWOPI),1.0
- AMCD((1.0-ATAN2(AIMAG(CDPlFR),REAL(CDPIFRI)/TWOPI .1.0
- ANCOiIE.0-ATAN2IAIMAG(CDP1IN),REAL(CDP1IN))/TWUPI),1.0)
- AMCD((1.0-ATAN2(AIMAG(COPODG),REAL(CDP10GI)/TWPI)e1.0)
= AMCO((1.0-ATAN2(AIMAG(CDP1
),REAL(CDPI
=
))/TWOPI),1.0)
VERSION 2.4
10/16/69
ZO1)
X1i)
P *
SUBROUTINE STAPR2 CALCULATES THE STEADY STATE CCNOITIONS IN THE
BOILING CPANNEL STARlING AT THE BOILING BOUNDARY AND MARCHING
DOWNSTREAM.
'NLY FOR UNII-ORM HEAT FLUX DISTRIBUTION.
SEGMENTS OF EQUAL TRANSIT TIME ARE CONSIDERED.
V()
Jul
oMI
111
:
C
C
C
JUALITY
C
: FLOWING SPECIFIC VOLUMES (FT3/LBM)
PROPERTY FUNCTI-NS
I
*
1,
IMAX-1
REUIRtO: ROLF, ROGF, MULF, MUGF, HFGF, HLF
PUL,
REAL
MUG, MULE, r4UJF, J
ZMAX, PIN, P0B, PEX, TIN, DTSUBR, DT
COMMON// D, QP, G, Z8,
CCMMON /STA20T/
B)60), C(60), X(60), ZO(60), PO(60),
1
UPDLFR1601, UPOLGR(60), OPAC160), V(60), J(60)
0M(60),
C
C
C
C
C
*********
THE
FREON
-
115 PROPERTY FUNCTIONS
ARGUMEAT CF RCLF, ROGF, MULF AND
MUGF
******
MUST BE ALOG(P)
=
01*X
.1033CPE 03 - .2657116E
- .830T981E -01*X*X*X
.3389485E -01*X*X
- .326T308E C1 + .9181820E 00 *X
EXP(
+ .3519715E-C2 *X*X )
I
.1064657 E 01 - .3255410 E 00*X
MULF(X) * EXP(
.1726090 E-02*N*X*X I
+ .6C64336 E-02*X*X
.1967795 E-01 + .7899068 E-02*X
MUGFIX) + .2141095 E-02*X*X*X
- .5630545 E-02*X*X
+ .1974262 E-04*X*X*X*X*X
2
- .4536185 E-C*X*X*X*X
RCLF(X)
-
I
ROGF(X)
ROLF
ROLF
ROGF
ROGF
MULF
ULF
MUGF
1
1
MUGF
MUGF
C
HFGF(P)
1
2
C
C
C
C
C
=*
.7058670
+ .4283483 E-01*P*P
+ .1250690 E-C4*P*P*P*P
E 02 - .9218673 E 00*P
- .1186139 E-02*P*P*P
HFGF
HFGF
HFGF
HLF MUST BE PROVIDED EXTERNALLY
G*(1.0-X)*QIMUL
= (-3.3256E-11)*FF(RELPF(X))*G*G*(1.0-X)*(1.C-X)/(RCL*D)
+ ROL)/144.0
DPDLGF(ALFA)
=-U(ROG-ROL)*ALFA
BGF)XALFA) = (-1.6628E-11)*((1.0-X)*(1.0-X)/((1.0-ALFAi*ROL) +
X*X/(ALFA*RCG))
XTTF(X) - (((1.0 -X)/X)**0.9)*((ROG/ROLI**0.5)*(MUL/PUG)**0.1
1
C
A - 0.7854 *0*0
QTP - QP/A
112
XTTI - XTTE
XITTE - XTTFIXEI
DPULF - FI2LF)O.5*XTTI+XTTe)l*DPDL-F(O.5*(X(I)+XF)I
UPOLFRII) - DPDLF/G**1.8
ALFAI - ALFAE
ALFAE = ALFAFIXTTE)
DPDLGR'(I) - DPDLGF(0.5*(ALFAI+ALFAE))
DPACI) = 8GFIXE
, ALFAE) - BGFIX(11, ALFAI)
DP = IDPULF + OPDLGRII))*DL + DPAC(I)*G*G
PEFIN * PI + UP
IFIPEFIN .LE.PEX)
PEFIN = PEX
P = ALOG(PEFIN)
VL * 1.0/FOLFIP)
VG
1.C/ROGF(P)
V(I+1) - VL + XE*(VG-VL)
CMI = ALOG(V(1+1)/V)I))/DT
IFIABS(PEFIN-PE/PEFIN - 0.002) 112,112,111
PC(1+1) - PEFIN
OMI)
*
C1I
ElI)
*EXP(0MI*DT)
Bl)
=
(EII)
-
DT)/CMI
FT21
B/HR-FT3
1.0)/UMI
X(1+1) - XE
J(I1+1) = G*VfI+1)
ZO(I+1) - ZC(I) + DZ
IF(ZO(I+1) - ZMAX) 9S9,998,998
998
* I + 1
RE TURN
999
CONTINUE
C
IMAX - 1000 IS USED AS A CJDE TO INDICATE THAT EXIT WAS NOT REACHED
IMAX - 1000
DTSKC = 360C.0 * DT
WRITE(6,1) 2C(60), UTSEC
FORMAT(40************* STAPR2 ***** 60 ITERATIONS COMPLETED WITHOU
I
REACHING ZMAX, Z0(60) -*, F10.5 /
OT(SEC)
* F10.5 //)
2
*
RETURN
997
IMAX - I
WRITE(6,2) IMAX, ZO(IMAX)
STAPR2 ******** A JUALITY LARGER THEN 1.0
2
, -WAS OBTAINED BEFURE REACHING ZMAX, 10(', 12, *1 -*,
I
F8.5 //l
2
RETURN
END
IMAX
IT
PRESSURE CROP FUNCTICNS
RELPF(XI
DPDLFF(X
DROP USING THE LOCKHART-MARTINELLI CORRELATION
P * ALOGtPE)
ROL = RCLF(P)
RUG - RlGrtP)
MUL - MULFIP)
MUG = MUGF(P)
RY
X, ZO, J ANE PC ARE DEFINED EOR I = 1, IMAX
014, B, E AND THE PRESSURE DROP TERMS ARE DEFINED FOR
V())I/HFGF(PBSI
nO 999 1 - 1,59
HI * HE
PI * Pol)
PEFIN * Pt
PE - PEFIN
El - (EXP(OMI*DT)
- 1.0)/041
DL - Jill
* El
HE a HI + QTP*DI/G
XE - (HE - HLF(PE)l/HFGF(PE)
IF(XE .CE. 1.0) GO TO 997
CALCULATE PRESSURE
(PSI/(LBM/HR-FT2)**2)
V
1.0/P0LFIP)
C
AUMBER CF SPACE POINTS
CM, 9,E : COEFFICIENTS DEFINED IN TEXT
VALUES
X :
20: POSI'TICN ALCNG THE CHANNEL (FT)
J: VELOCITIES OF THE CENTER OF VOLUME (FT/HR)
VALUES (PSIA)
PC: PRESSURE
DPDLR :
FRICTIONAL PRESSURE URUP SLOPE DIVIDED BY G**1.8
(PSI/FT) / (LKM/HR-FT2)**1.8
OPDLGR : GRAVIATATIONAL PRESSURE DROP SLOPE (PSI/FT)
G**2
: ACCELERATION PRESSURE DROP VALUES CIVIDED
OPAC
- ZoB
* 0.0
ALCC(PBRI
*
* G * V(1
TP * t.0/ROGF(P) T
HE * HLF(PAB)
XTTE * 1.OE+30
ALFAF
1.0E-10
REQUIRED INPUT I
QP : LINEAR HEAT RATE (BTU/HR-FT)
G : MASS ILUX (LBM/HR-FT?)
LRB
POSITION 0F THE ROILING BOUNDARY (FT)
ZMAX : MAX VALUE CF COORDINATE ALONG THE CHANNEL TO BE
REACHED (FT1
P88 : PRESSURE AT THE BOILING BOUNDARY (PSIA)
PEX : LIMITING EXIT PRESSURt VALUE (PSIA)
UT : TRANSIT TIME INCREMENT (HR)
OUTPUT
IMAX :
v PB8
Po(1)
SUBRCUTINE STAPR2(IMAX)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
FORMAT(*0**********
SUBROUTINE DYNPRL (KMAX)
11/07/69
VFRSION 1.3
SUbROUTINE UYNPRI CALCULATES THE PRESSURE DROP IN THE SINGLE-PHASE
REGION IA A NCK-LINEAR FASHION
C
C
C
C
C
C
C
=1,KMAX
ARE
Z11,K) ANF C(1,K) (-UR K
PI1,K) FOR K -1,KMAX IS THE OUTPUT
TO THIS SUBROUTINE
INPUT
REAL MULB, MULTF, KLF, 4ULW
GO, ZBB, ZMAX, PI , P88, PEX, TIN, DTSUBB, OT
CCMMON // D,
CUM*CN /UYNPH/ Z(40,120),
ZOT140,120), G(40,120), P140,120)
QP,
C
C
C
PROPERTY FUNCTICNS
REQUIRED :
ROLTF,
KLF
MULTF,
TIlT) = 1.0 / (T + 459.61
MULTF(T)= EXPI - .5483695 E 01 + .5585921 E 04*TIIT)
+ .3577018 E 09*TI(TI**31
- .1950534 E C7*TIIT)**2
1
.1035453 E' 03 - .7105255 E-01*T - .6448694E-04*T*T
ROLTF(T) KLF(T) - 0.0475 - 6.76667E-05* T
MULTF2/2
ROLTFt/1
KLF
1/1
=
JLI(120,
021(120)
REAL
JI(12C),
ZMAX, P1 , PBS, PEX, TIN, CTSURB, OT
COMMON// D,
GLZe8,
COMMON /DYNPR/ Z(40,1201, aZDT(40,120l, G(40,120), P(4C,120)
CCMMON /STA20T/ CM(60), 6(601, E(60), X(601, ZO(60), PO(601,
1
OPOLFR(60), DPDLGR(60), UPAC(60I, V(601, AJO(601
1MAXI a IMAX - 1
KMAXI - KMAX
DTINV - 1.0/0T
OTINCO - 1.0/((36CO.0*3600.0*32.17*144.0I*DTI
C
C
C
89
C
C
C
CALCULATE THE VOLUMtTRIC
C
C
C
99
C
C
C
FLUX DENSITIES
AT THE 8.B.
.
. .
.
. .
.
00 89 K = 1,KMAX
JI(K)
G(1,IK) * VIl1
MAJCR LOOP
00
1
Z(1,K),
C
C
P802 = (DPDLFR*GOP*1.
+ CPDLGR)*ZBB + PI
IF(A8S(Pd82-PBBf/PBB .GE. 0.02) WRITE(6,11 PB11,P882
FCRMATIO0************STEADY STATE SINGLE PHASE PRESSURE DROP AS
*
,'CALCULATED NY DYNPRI DOES NOT AGREE WITH SUBCBB VALUE :/
2
30X, 'P88(SUBCBR) =', F10.4, 1OX, 'PBBIDYNPRI) =0, F10.4,
3
* PSIA' //)
KMAXI - KMAX -1
DPI-ROI - -T.OIE-1O*MULTF(TIN)**0.2/ROLIN
PIP = P1 - DPFRO1*GO**1.R
U(1,K),
K-1,KMAX : VALUES AT THE
P(I,K), DiT(1,K),
STARTING POINT (USUALLY THE eg), AND
E(I), B(11,
PDLFR(I), DPDLGR(Il, DPAC( I1,
OM(I),
FUR I = 1, IMAX - 1
V(I1 FOR I
1, IMAX AND OT. : REFERENCE STEADY-STATE
PARAMtTERS
INPUT :
QP,
MULTFI/2
C
TAV - TIN + 0.5*DTSUBB
RCL - ROLTFITAV)
ROLIN - ROLTF(TIN)
MULB - PULTF(TAVI
DTINCC - 1.0/((3600.C*3600.0*32.171144.01*DT)
DTFILM = (282.0*1.291667)*)P*MULB**0.4/IGO**0.8*KLF(TAV)**0.61
TWAV = TAV + DTFILM
MULW - MLLTFITWAVI
RE - GO*C/MULB
DPOLGR = -RCL/144.0
UPDLFR - I-3.3256E-11FFRE)*(G0**0.2*(MUL/MULSI**0.14/RCL*D
DYNPR2 (IMAX, KMAX, KMAX21
11/07/69
VERSION 1.5
SUBROUTINE DYNPR2 CALCULATES TIMt UEPENDENT PRESSURE DROP IN A
TWO-PHASE CIANNtL ACCJRDING TO THE ENTHALPY TRAJECTCRY MODEL
SUBACUTINE
C
C
C
C
C
C
C
C
C
C
C
999 1 *1,IMAX1
DETERMINE RELATIVE
DO 99 K
JZIRI *
VEL3CITIES AT INLET
= 1,KMAXI
Jt(K)
KMAXI = KPAXI
DZCT(I,K)
-
FLUX,
CAtCULATE-RASS
-
VELOCITIES
AND COURDINATES OF MESH POINTS
1
C
C
99
DO 99 K = 1,KMAX1
G18 = GI1,K)**1.8
DGOT - (G(I1,K+1-GTI,Kt*TINC1D
OPTIFI - DPFR01*G18 - DGDT*2.3
DP - (DPDLFR*G18 + OPDLGR - OGDTI*Z(I,KI
PI1,K) = PIP + UP
RETURN
kND
DO 98 K = 1,KMAXI
SIK - (JZI(K+1I - JZI(KI) * OTINV
DZI(K) aE It) * JZItK) + B(I) * SIK
+
DPO1FI
C
C
C
C
98
C
C
C
CALCULATE VALUES FOR NEXT SPACE STEP . .
. .
. .
.
. .
, .
.
. .
.
. .
.
. .
. .
.
. . .
JIfK) - JI(K+1I + CMIII * DZIfK)
JIK) HERE IS REOEFINED AS J(I+.1,K)
G(I+1,K) - JIIKI / V(I+1)
Z(I+1,Kl = Z(I,K+1l + DZI(K)
DZDT(I+1,K) = DZOT(IK+1L) + E(I)*SIK
CALCULATE PRESSURE DRCP AND PRESSURE .
KMAX2
GAVIKI
=
KPAXI - 1
= (G(1+1,1)
+
. .
.
Gil,2))*O.5
C
00 97 K =1,KMAX2
GAVIK = GAVIKI
GAVIKI - (G(I+1,K+1) + G(I,K+2)I*0.5
DPIx = (OPOLFR(I)*(AVIK**1.8 + DPDLGR(II)
1
*DZIK) + DPAC(I)*GAVIK**2
97
F(I+1,IK
- P(I,K+IJ + DPIK
999 CONTMNUE
RETURN END
-
i6AVIK1-GAVIK)*DTIACC)
NM.
SLBROLTINE PRTPL3(N3,t!,
VERSICN4
YSC)
KX,JX,
NL,NS,
12
14
15
16
18
20
OF
A SUBROUTINE SUPPLIED BY THE
PRTPL3 IS A MOCIFIEC VERSION
PRINTER PLOTS ARE PRODUCED
INFORMATICN PROCESSING CENTER AT MIT.
AND THE EXTREMA OF TFE GIVEN FUNCTIUNS ARE DETERMINEO.
C
C
C
B(KX,JXI
DIPEASICA CUT(10I),YPR(I11I, IANG(9),A( 480),
IANG/.,G'3,'4','5','6','7','8','9'/
INTEGER CUT,
CCMlON/PLCTCT/ XMAXJ(9), YMAXJ(9), XMINJ(9), YMINJ(9), YRANJ(9)
C
VER. 4.1
1 FORMAT(II,
2 FCRMATIIH,
60X,7H CHART ,13,/)
F11.4 , 5X, 101AX, 112)
3 FORMAT(IH, 117X, 112)
4 FORMAT('0PRTPL3
MIN ORDINATE =',
F12.5, * AT ABSCISSA
F12.5, *
(CURVE NO-,
12, * ', IX, Al /
1
2
11X, 'MAX ORDINATE
F',
12.5, I AT ABSCISSA
F12.5,
30X, 'ORDINATE RANCE =', F12.5)
3
7 FORMAT( 1H 16X,101H.
.)
.
.
.
.
.
1
8 FORMAT(H ,9X,11F10.4)
IF((YSC
IADV
C
C
C
DETERMINE EXTREMA
42
41
43
40
45
C
50
55
57
1
OFEACH
56
. .
CURVE.
.
. .
. .
.
. .
. .
.
. .
. . . . . . .
A(I)=(K,J)
DO 80 11,NLL
F=I-1
XPRTXO+F*XSCAL
IFIAIL)-XPR-XSCAL*0.5) 50,50,70
DO 55 1X=1,101
CUT(IX)BeLANK
CONTINUE
00 60 J=1,MY
LL=L+J*N
JP=l(ALL)-YMIN)/YSCAL)+I1.0
OUT(JP)=IANC(J)
CONTINUE
IFIL.EQ.N) GO TO 61
L=L+1
57,57,61
CONTINUE
WRITE(6,2)XPR,IOUT( IZ), 1Z=1,101),
GO TO 8C
WRITE16,3) I
CONTINUE
WR
ITE(6,7)
YPR(1)YMIN
DO 90 KN=1,10
YPR(KN+1)=YPR(KN)+YSCAL*10.0
WRITE46,8)(YPR(IP),IP=1,11)
IF(A)L)-XPR-XSCAL*0.5)
61
70
80
90
C
C
.
. . . . . . . . . . . . .
IF(YSC .LE. 0.0) GO TC 91
YMAX=-1.0E75
YIN=1 .CE75
DO 49 J = 1,MY
IF(YMAXJ(J) .GT. YMAX) YMAX = YMAXJJ)
IF(YMINJIJ) .LT. YMIN) YMIN = YMINJ(J)
49 CONTINUE
YRANGE = YMAX - YMIN
1=1
DU 39 J=1,M
DO 39 K=1,N
39 1=1+1
ALL=NL
IF(NS) 16, 16, 10
10 DC 15 I=1,N
DO 14 J=1,N
IF(A(I)-A(J)l 14, 14, 11
11 L=I-N
LL=J-A
DO 12 K=1,M
L=L+N
LL=LL+N
F=A(L)
60
. .
DC 45 J - 2,M
YMIN=1.CE75
YMAX*-1.OE75
DO 40 K = 1,N
IF(8(KJ) - YMAX) 41,41,42
YMAX = B(KJ)
XMAX = B(X,1)
IF(B(K,J) - YMIN) 43,40,40
YMIN - B)I,J)
XMIN = B(K,1)
CONTINUE
XMINJ(J-1) = XMIN
XMAXJIJ-1) = XMAX
YMINJ(J-1) = YMIN
YMAXJ(J-1) = YMAX
YRANJ(J-1) = YMAX - YMIN
PLOT THE CATA . .
YSCAL=0.01*YSC
L=1
=,
IADV = 0
IF(Y$C .GT. 0.0)
WRITE(6,1) IADV, NO
MY = M - 1
A(L)=A(LL)
AILL)=F
CONTINUE
CCNTINUE
IF(NLL) 2C, 18, 20
NLL-50
WRITE 6,7)
BLANK=0
XSCAL=(AIN)-A(1))/(FLCAT(NLL-1)I
YSCAL = C.01 * YRANGE
IF(YSCAL .EQ. E.0) YSCAL = 1.0
.GT. YRANGE) .ANU. (YRANGE .GT. YSC/10.0))
X8=A(1)
. . *.
I
WRITE THE VALUES OF TIHE MINIMUM AND MAXIMUM OF EACH CURVE
91 DO 92 J = lPY
92 WRITE(6,4) YMINJ(J),XMINJ(J),J,IANu(J),YMAXJ(J),XMAXJJ),YRANJIJ)
RETURN
END
.00
011111'
PRESSURE
RELIEF
VALVE
~R'
SER
PARTIAL
(W)WATTMETER
PRESSUREGAUGE- BOURDON
PRESSURETRANSDUCER
g THERMOCOUPLE
BALL VALVE UNLESS NOTED
ST'D ASME ORIFICEFLANGE
AND PLATE
VENTURI FLO*METER
MANOMETER
-
END
VIEW
CLEA PLASTIC COUPLING
1/2 THICK
7 GROUPSOF THREE PARALLEL
0430" ID x 0.510" OD GLASS TUBES
VARIABLE
220v TRANSFORMERFOR
60- EACH SET OF THREE
TUBES IN PARALLEL
3 PARALLEL TIE RODS (NOT SHOWN)
AND METAL COUPLING SPACERS
GID CAGE IN WHCH THE
FORM A
TEST SECTIONS AM MOUNTED
3 STATIC TEST FLUID PRESSUREAND
,13 BULK FLUID TEMPERATUREMEASURING
STATIONS AT EACH COUPLING
ET
IMAT
IDENTICALPROVISION FOR MEASUREMENTS
FOR EACH TEST
15.5"INTERVALS
SECTION
-fo-THREE
TEST SECTIONS SIMILAR
COPPER INLET SECTION(UNHEATED)
NOMINALSIZE, INCHES,ASTM SCH 40
NOMINAL SIZE , INCHES,ASTM TYPE L
LENGTH, FEET
SYSTEM HAS NO INVERTED"U-BENDS"
FOR GAS POCKETS
P
T
L
FIG 2.1
SCHEMATIC OF APPARATUS,
WITH OVERALL DIMENSIONS
-
9042
1
---
7.750
Lh
-- - 9.302 ft.-
.500
FIG. 2.2 CODING OF LOOP EQUIPMENT AND TEST SECTION DIMENSIONS
1,
0-RING (EXAGGERATED)
COUPLING PLATE (2) - CLEAR PLASTIC
0.5". -- - _
COUPLING - CLEAR PLASTIC
0.5'"'
0.5"1
0.5"
GLASS TUBE WITH ELECTRICAL
RESISTANCE HEAT FILM, 0.0430" I.D.
iv13.0"
0.510" 0.D., 15" LENGTH
TIE RODS (NOT SHOWN)
PROVIDE STRUCTURAL
SUPPORT AND ALIGNMENT
FLUID STATIC PRESSURE
TAP 0.042" DIA.
BULK FLOW
THERMOCOUPLE
0.062" DIA,
WIRE BRAID POWER
LEAD
FIG. 2.3 CROSS-SECTIONAL DETAIL OF TEST LENGTH AND COUPLING
VENTURI
.
.0A
17 l
it.%PVS
1 06
2kL~
TRANSDUCER
BRIDGE
ADAPTER
DIRECT
RITING
PSCILLOGRAPH
PREAMPLIFIER
CALIBRATION
CIRCUIT
l
5k+l
l
REGULATED
POWER SUPPLY
i5 V; 5o v*A
All operational amplifiers
made by Philbrick
INTEGRATOR
FIG. 2.4 FLOW MEASURING INSTRUMENTATION
z
-
lug Flow Model
0.1 Zuber and Findlay
/.
Homogeneous Model
LM
>
Freon -113 at 20 psia
G- 3.5 x I10 Ibm /hr ft"
G-IOxlO lbm/hrft"
D- 0.43 in.
L ock hart - Mart inellIi
l
0.01-
001
0001
0.0001
QUALITY, x
FIG. 3.1
COMPARISON
OF VOID FRACTION CORRELATIONS
0.1
1.0
100
FREON-i 13
80
p = 15 psia
I'S MODEL
11750 Btu/hrft
p = 22.5 psia
2
D = 0.43 in.
0
z
LAL
0
z
0
0
C-)
Zn
STAUB
= 2350
LEVY'S MODEL
q" = 2350 Btu/hrft
2
-20
2
MASS FLUX,
5
G
10
[Ibm/hrft 2 ]xiof
FIG. 3.2 POINT OF NET VAPOR GENERATION AS PREDICTED BY TWO MODELS
API ex
8
[psi]
40
60
100
80
INLET TEWERATURE,
T0
120
[*F]
FIG. 3.3 EFFECT OF THE INLET TEMPERATURE ON VARIOUS PARAMETERS
140
zbb
8
xex.
7
.8 -
POSITION OF THE BB, zbb
[ft]
/
18 1-7
/
[psia]
APlex,
6
8
5
ATsubb
[psi]
TOTAL PRESSURE DROP, APlex
50 r
7
4
66
40 .
3
/
30
5
20
4
L
SUBCOOLING, AT subb
16 1-2
/
pex = 14.7 psia
T1= 90 *F
EXIT QUALITY, xex
3
q'= 400 W/TL
Uniform heat flux distribution
0
No subcooled boiling
0
PRESSURE AT THE BB, pbb
1000
MASS FLOW RATE, w
1500
[lbm/hr]
FIG. 3.4 EFFECT OF THE MASS FLOW RATE ON VARIOUS PARAMETERS
2000
WAM0WlMNMM
Wll0lhlli,
I
I
06
0.5
2'
1.0
MASS FLUX, G [
FIG. 3.5
5
Ibm/hr ft]
RELATIVE IMPORTANCE OF THE FRICTION, GRAVITY AND
ACCELERATION
TERMS.
CONTOUR MAP AT 792 Btu/hrft
10
I
I
I
I
I
I
I
I
UNIFORM (1150
COSINE (115.2 *F'
,a
UNIFOR
0
13
(96.0 *F
COSINE (94.8 *F)
c:)
Lu
LU
c-
UNIFORM (115
*F)
RUN DISTR. T
0
a 4A
* 8
-
I
I
UNIF. 95
COS. 95
o 10 UNIF.115
//
* 14
COS. 115
LINES ARE 'PRESDROP' PREDICTIONS
500
1000
I
I
1500
2000
MASS FLOW RATE, w [lbm/hr]
FIG. 4.1 PREDICTED AND MEASURED TOTAL PRESSURE DROP AT
q" = 4700 Btu/hrft 2
,
2500
14
0
AV
11, 12, 16 UNIFORM
115
115
17
ROOFTOP
115
UNIF
OO
6,19,22
UNIFORM
95
UNIFORM(15.AF )
*
9
COSINE
95
*
13
A
ROOFTOP (115.2OF
COSINE
*
TO
(115.2 *
COSINE (114 5 *F
12
0
-
0.0
. e
0
o
100
w
0
UNIFORM (94.6 *F)
0
I,00
--
COSINE (94.8 *F)
o
00 -
ISOTHERMAL (95.20 F)
LINES ARE'PRESDROP' PREDICTION!
4
0
1000
MASS FLOW RATE,
FIG. 4.2
3000
2000
w
4000
[Ibm/hr]
PREDICTED AND MEASURED TOTAL PRESSURE DROP AT q"= 9400 Btu/hrft?
A RUN 15
O RUN 7
UNIFORM HEAT FLUX DISTRIBUTION
o RUN 5
To =15.2 0 F
Pc z 30.40 in.
In
a- 12 -
wA
a.
To r=94.6* F
Pc = 30.24 in.Hg
0
a
o
a. 0-
T~9.
-
0
00T
8
-3
=95.2* F
4
LINES AR E'PRESDROP' PR EDI CTIONS
0
1000
2000
3000
FLOW RATE, w (Ibm/hr)
FIG.43 PREDICTED AND MEASURED TOTAL PRESSURE DROPS AT q"= 14100 Btu/hrtt2
4.48 fit/s
ex =-0.036
STAUB'S NVG CORRELATION
LEVY'S NVG CORRELATION
2.79
-0.008
C.
44-
2.18
O.
0 -0.013v
BB
NVG
00
I-3
1. 64 -...
0.044
B
NVG
1.250
0.084
BB
oTo
RUN NO 8
= 94.8 *F
0
NVGRB
0
B
1
2
3
4
5
6
7
STATION ALONG THE CHANNEL
FIG. 4.4 PRESSURE PROFILES AT 4700 Btu/hrft 2, COSINE HEAT FLUX DISTRIBUTION
8 ex
2
(o)
5.73
- 0.010
.-. , 4.48
13
0.011
(0)
3.70 V
0.030
0O
z
0
Ies
>
L.
0
1.74 o
0.157
0
it
RUN NO 6
W2
T.
94.6 *F
0.95
I
2
3
4
5
6
7
STATION ALONG THE CHANNEL
FIG. 4.5
PRESSURE PROFILES AT 9400 Btu/hr ft, UNiFORM
HEAT FLUX DISTRIBUTION
8BOx
RUN NO. 15
EFFECT OF NVG CORRELATION
T*' 115.2*F
8k
0.
N
0
2r
6 1-
0.
O
I--
Staub
IIII
I11II1
11
4
7
5
STATION NO. ALONG THE CHANNEL
2
FIG.4.6 PRESSURE PROFILES AT 14,100 Btu/hrft, UNIFORM
HEAT FLUX DISTRIBUTION
8ex
1
I
C:
I
I
I
I
I
o A
o
U
1.0
uj
A
O
oy
0 o
AA
v
O
o v
vT A
VA
6
v.
a-
L
RUNS D4 TO D21
AVERAGE POWER
o 100 W/TL
S0.8 -
V 200
OPEN POINTS : UNIFORM HEAT FLUX DISTRIBUTION
A 300
O 400
CLOSED POINTS : COSINE HEAT FLUX DISTRIBUTION
o
0.1
0.2
0.3
EXIT QUALITY,
0.4
0.5
500
0.6
4-)~
0
0f
oL
0
I'D c:)
r-
4r-
C\j L0
r--
C\J
LfA
C\J
I VERAGE VALUES AND
STANDARD DEVIATIONS
0.7
xe
FIG. 4.7 ACCURACY OF THE PRESSURE DROP PREDICTIONS FOR ALL THRESHOLD AND TRANSITION POINTS
lull
11
il1,,,,,llolmini
I
I
6
o FIRST CYCLE
A SECOND CYCLE
O
THIRD CYCLE
o FOURTH CYCLE
I
4
MAXIMUM FLOW REACHED
CLOSED POINTS INDICATE
DECREASING FLOW
PRESSURE AT STATION I
2
N
0
w
co
w O)
0r
PRESSURE AT STATION 6
0.4-
0
~
I
I
I
2
4
6
INLET VELOCITY, V,
FIG. 4A
PRESSURE
0
8
ft/s
DROP HYSTERESIS
10
150
Point A
o = 3.93 ft/s
Tsat
0.027
ex
130 -
Tcal*
110 U0
LLJ
LU
90
1
2
3
4
5
6
7
8ex
LU
S 150
Tsat
LU
130-
0
110 -
Tcal
Point B
3.72 ft/s
= 0.033
Vo
T
x
90
1
1
2
3
4
I
I
I
al 1
5
6
7
8ex
1
STATION NO. ALONG THE CHANNEL
Tsat ; Saturation temperature based on measured pressure
T
Calculated single-phase temperature
cal
T
Measured temperature (Low temperature reading at
Station 8 is due to a thermocouple error)
FIG. 4.9 TEMPERATURE PROFILES FOR POINTS A AND B OF FIG. 4.8
WBMV
Mwolll
11
1 .0__
M1911ill
_
___
_
jQ*(z'jbj)
0.1
2 IH*(z,jw)I
0.001
C~j
I
0.0001
I
100
10
1
0.1
C.J
C~j
1000
2ir
argQ*(z,jw)
3
S 21
di
1
di
2
2
0.1
100
10
1
argH*(z,jw)
1000
Period, T [s]
Uniform heat flux distribution; V0 = 1 ft/s, T = 100 *F;
q = 800 Bt.u/hrft; a = 0.8.
Curve 2 is the high frequency approximation.
2'-k = 16.4 s;
2
7TTh
= 34.9 s;
2Tc = 47.9 s.
FIG. 5.2 FLOW-TO-LOCAL-ENTHALPY AND FLOW-TO-HEAT-FLUX TRANSFER
FUNCTIONS AT
z = 2 ft.
Im
Effect of T [s]
4-)
cnj
Effect of q'
rft]I
-[Btu/h
Reference point
zbb = 3.0 ft
q = 800 Btu/hrft
G = 3.5x10 5 ibm
Effect of G
[105 lbm/hrft2]
hr ft2
T
6 s
5/
I.e
Effect of z0
[ft]
[Btu/1 bm]
FIG. 5.4 PARAMETRIC STUDY OF THE ENTHALPY PERTURBATION
AT THE BOILING BOUNDARY, 6 hob
0.08
I
I
I
I
I
|
I
I|
0.06
* 0.04-
0.02
0.000
0
I
r~3
22
.--
3,
C\j39
BB
0
2
4
6
Length along the channel, z [ft]
Experimental Point D20-192;
-r= 11.70 s;
X = 13.26 ft.
1:
2:
3:
4:
V = 1.133 ft/s;
0
Wall heat storage and cold spots taken into account
Only wall heat storage taken into account; uniform heat input
High frequency approximation, Eq. (5.38)
Low frequency approximation. To get the amplitudes divide the
amplitudes of
3 by (1 - a)
Heated and cold spots are shown by black and white line between the graphs.
FIG. 5.6 FLOW-TO-LOCAL-ENTHALPY TRANSFER FUNCTION ALONG THE
CHANNEL. ZERO ORDER OSCILLATION.
0.015 -1
2
0.010-
3
N3
0.005
B
0.000
i
2
i
'0,
2
N
344
s.-
C\j
BB
0
3
2
1
Length along the channel, z
Experimental Point D19-184;
[ft]
T = 4.46 s;
V0 = 0.410 ft/s;
X = 1.83 ft.
1 , 2 , 3 , and 4
See Fig. 5.6
FIG. 5.7 FLOW-TO-LOCAL-ENTHALPY TRANSFER FUNCTION ALONG
THE CHANNEL. FIRST ORDER OSCILLATION.
j11IM110
0.005
0.004
0.003
0.002
0.001
0.000
M
M
is
2w
1
2
3
Length along the channel, z
4
5
[ft]
T = 3.90 s; V0 = 0.291 ft/s;
X = 1.135 ft.
Experimental Point D18A-196;
1 , 2 , 3 , and
4
See Fig. 5.6
FIG. 5.8 FLOW-TO-LOCAL-ENTHALPY TRANSFER FUNCTION ALONG
THE CHANNEL. FOURTH ORDER OSCILLATION.
,..,.,,.....111..
u
mm
illllllllll1llllllllllllllI1llllll
0.08
0.04
0.00
3
2
. 0
1
2
3
4
Length along the channel , z
5
6
[ft]
T = 11.70 s;
X = 13.26 ft.
Experimental Point D20-192;
V0 = 1.133 ft/s;
Cosine heat flux distribution shown on top.
FIG. 5.9 COMPARISON OF THE-FLOW-TO-LOCAL-ENTHALPY TRANSFER
FUNCTIONS FOR UNIFORM AND COSINE HEAT FLUX
DISTRIBUTIONS. ZERO ORDER OSCILLATION.
"11411
0.004
0.002
0.000
0
1
2
3
4
Length along the channel, z
Experimental Point D18A-196;
T-
5
6
Eft]
= 3.90 s;
V0 = 0.291 ft/s;
A = 1.135 ft.
Cosine heat flux distribution shown on top.
FIG. 5.10 COMPARISON OF THE FLOW-TO-LOCAL-ENTHALPY TRANSFER
FUNCTIONS FOR UNIFORM AND COSINE HEAT FLUX
DISTRIBUTIONS.
FOURTH ORDER OSCILLATION.
0.004
0.002
0.000
27r
3
Tr
2
Tr
2
10
8
6
4
Length along the channel, z [ft]
The cosine distribution is simulated by N = 7, 14, and 28
segments. Two corresponding heat flux distributions are shown
at the top.
0
2
FIG. 5.11 EFFECT OF THE CHANNEL SEGMENTATION ON THE FLOW-TO-LOCAL
ENTHALPY TRANSFER FUNCTION.
- , , jP6
0
0.25
0.5
0.75
DIMENSIONLESS LENGTH , z
1.0
0
0.25
DIMENSIONLESS
FIG. 5.12 ENTHALPY PERTURBATIONS ALONG THE
CHANNEL, 8V/Vo0 O.I
0.5
TIME,
075
t*
0.5
0.25
DIMENSIONLESS LENGTH,
075
z*
1.0
0
0.5
0.25
DIMENSIONLESS TIME,
FIG. 5.13 ENTHALPY PERTURBATIONS ALONG THE CHANNEL,
8V/Vo= I
0.75
t*
mmhuuuumummin
Flat Pressure Profile
minmmumm*iui
~im.im~
Static Pressure Effects
FIG. 5.14 EFFECT OF THE
PRESSURE ON THE MOVEMENT
OF THE BOILING BOUNDARY
6w
Re
friction
6Szbb
B. 6h in Second Quadrant
A. 6h in First Quadrant
FIG. 5.15 EFFECT OF THE DYNAMIC PRESSURE VARIATIONS ON THE MOVEMENT
OF THE BOILING BOUNDARY
60
<
D18A
-
40
D17
unstable
_
D10A
--
_
+
20
EXPERIMENTAL POINT
THRESHOLD POINT
I TRANSITION POINT
o INTERPOLATED THRESHOLD
0,1,
ORDER OF OSCILLATIOt
D9
etc
D6
D4
D21
RUN NUMBER
2TR2B
0
0.2
0.6
0.4
0.8
q/w h
0
FIG. 7.1 STABILITY MAP AT 100 W/TL, UNIFORM HEAT FLUX DISTRIBUTION
1.0
lill
022
N~~
m milii ,
.'i,,WimuI
0.4
l
0.6
0.8
q/w hfg
FIG. 7.2 STABILITY MAP AT 200 W/TL, UNIFORM HEAT FLUX DISTRIBUTION
60
40
H-
40
20
0
0.2
0.6
0.4
0.8
q/w0 hfg
FIG. 7.3 STABILITY MAP AT 300 W/TL, UNIFORM HEAT FLUX DISTRIBUTION
1.0
1 I1li 1l haia
u u'lll
liilll
lll1ll
0.2
0.4
0.6
ill
lllin llull
i
lll
lll0it
ll
IIMI.l..
..i
liiiiii
ll4
l i
0.8
q/w0 hq
FIG. 7.4 STABILITY MAP AT 100 W/TL AVE., COSINE HEAT FLUX DISTRIBUTION
t
Ill
'
ill
blii
i
i
il
A i
il
II
I
D21
STABLE REGION
D23
40 I-
D22
+0
\ D11
20
F
SYMBOLS EXPLAINED IN FIG. 7.1
0.2
0.6
0.4
0.8
1
q/w0 hfg
FIG. 7.5 STABILITY MAP AT 200 W/TL AVE., COSINE HEAT FLUX DISTRIBUTION
W**mp"wWW
9.001*0.0
Run TR8, 300 W/TL, unif. &Tsubb=6 5 .40F
we
Fundamental
mode
3
30s
Third-order
oscillation
fwadt
wI
___Is
~'
AO
[lbm/hrj
00
0
_
-4-
B Run Di0A
200 W/TL
unif.
Points 93
and 93A
Run D16
200 W/TL
unif.
Points 20
Idiv = I
and 21
FIG. 7.6 RECORDINGS OF OSCILLATIONS - TRANSITION POINTS
80
\~
/
-/
0
300 W/ TL
o/
O
G/
60
0/
A
~
(200)
c13
0
u>
0
co~40
(200)
U)
/
POWER LEVEL
/
A 100 W/TL
0 200
/
O
300
V 400
0 500
A
0
A
380
-
-
q / Wohfg
Open points: First occurrence of oscillation, zero order.
Closed points: First occurrence of oscillation , higher-mode
(Number of tails denotes order of oscillation).
Points with a bar: Transitions to fundamental model.
Crowley, Deane and Gouse's boundaries also shown.
OSCILLATION BOUNDARIES
FIG. 7.7 FIRST OCCURRENCE OF INSTABILITIES AND FUNDAMENTAL MODE
M n
-. 01111
11llililiiiillmlM
m
o
lM
o
lo
ll
I'
llI
l 1,, oI,t
iR
IIldia
n
i
i
,
,
ia
01111
- -
.,
,l
I
100
m500
CROWLEY,
GOUSE
DEANE AND
CLOSED POINTS DENOTE
TRANSITIONS TO FUNDAMENTAL MODE
80 I-
680 W /TL
600
r__0
\.
\
\
STABLE REGION
\.
N
380
C)
I
C)
C..)
I
-LJ
*
*
I
I
I.
I
I
*1 )
I
I
~-
I
I
I
a
INLET VELOCITY,
a
V0 [ft/s]
FIG. 7.8 STABILITY BOUNDARIES FOR UNIFORM POWER
DISTRIBUTION, ZERO ORDER POINTS ONLY
100
400 W/TL
300
80
I
\
'
20
660
200
STABLE
REGION
I
-J
C-)
I
100
//
'I
20
I
/
/
I
\I'
%
/
------
/
I'
I
POWER DISTRIBUTION
COSINE
------ UNIFORM
I
INLET VELOCITY,
V0 [ft/s]
FIG. 7.9 STABILITY BOUNDARIES FOR COSINE POWER
DISTRIBUTION - ZERO ORDER POINTS ONLY
ho
06
A0
A0
A
9
0.4 -A
0200
UNSTABLE
REGION
AA
oil
POWER LEVEL
A
c3
0.2
(C-D-G) 500
(c-D-G) 600
I
II
0
0.2
200
Closed points indicate transitions
higher zero-order curve at 200 l
(
see Fig. 7.2)
I
0.6
0.4
W/TL
0 300
v 400
C0 500
380
CDG38050
(c-D-G) 680
(--
0.8
q/w0 - hfg
FIG. 7.10 STABILITY MAP IN THE DIMENSIONLESS ENTHALPY-PRESSURE
DROP PLANE - ZERO ORDER POINTS ONLY
1.0
20
40
SUBCOOLING, ATsubb
60
80
[*F]
Open points denote transitions. Tails indicate order of oscillation.
Crosses mark decay of oscillations.
FIG. 7.11 PERIOD OF THE OSCILLATIONS AT THE THRESHOLD OF STABILITY ALL ORDERS AT 200 W/TL
12
8 -
0v
Lii
4
-
POWER LEVEL
A
Open points denote
transitions
1
60
0
0
20
40
SUBCOOLING,
ATsubb
A
100 W/TL
m
200
*
V
300
400
*
500
1
80
[*F]
FIG. 7.12 PERIOD OF THE OSCILLATION AT THE THRESHOLD OF STABILITY ZERO ORDER POINTS ONLY - UNIFORM HEAT FLUX DISTRIBUTION
'10
TO00
----
UNIFORM HEAT FLUX
DISTRIBUTION
81-
.-
400
-V
POWER LEVEL
41*
*
V
200 W/TL
300
400
Open points denote
transitions
SUBCOOLING,
AT subb
[OF]
FIG. 7.13 PERIOD OF THE OSCILLATION AT THE THRESHOLD OF STABILITY ZERO ORDER POINTS ONLY - COSINE HEAT FLUX DISTRIBUTION
m
ATsubb
[OF]
0 F]
T = 3
40-
2.5
20
0.4
0.8
0.6
q
FIG. 7.14 PERIOD OF THE OSCILLATION WITHIN THE
UNSTABLE REGION (CONTOUR MAP)
1.0
1.0
W
uniform
0.5
exact cosine
27r
resulting
uniform
difference
in delay
3
3
0
exact cosine
SUBCOOL ING
0
0.25
0.50
0.75
1.0
Relatiye Position of the BB along the channel, z'bb/L
FIG. 7.15 ENTHALPY AND SINGLE-PHASE DELAYS - UNIFORM VERSUS COSINE HEAT FLUX DISTRIBUTION
Run TRiA
300 W/TL,unif.
Run TR6
300 W/TL,unif.
ATsubb=3 4 .30 F
111we
I s - -0
Is
9
ATsubb= .3-5.40F
Run D19
Point 4
Run D18
Point 17
200 W/TL,unif.
AT uo=69.3*F
S ULI
F
Run E02
Point 6
psi
Run D21
Psnt 2 0
30W/TL,COS
Tsubb= 76. 2*F
AT
Wo
[
lbm/hr]
FIG. 7.16 TYPICAL OSCILLATION RECORDINGS
100
CORRECTED ORDER OF OSCILLATION
............ STABLE
THIRD-ORDER
====
--
SECOND-ORDER
FIRST-ORDER
LL,
200 W/TL
800 1
80
0-
---
-
6
E03 (all')
60
)E04
(-I
~~
60
-----
TL)
E-0TL
(al)0
-- BOUNDARIES OF HIGHER-ORDER
2TL
E05~
REGIONS FROM FIG. 7.2
/
40
0
40
E06
(-3
TL
0.2
0.6
0.4
CORRECTED
q*/w
0.8
- h
FIG. 7.17 STABILITY MAP IN THE ENTHALPY-SUBCOOLING PLANE - SERIES E EXPERIMENTS
1.0
wo
A
1.0
-0
CORRECTED ORDER OF OSCILLATION
...........
STABLE
THIRD-ORDER
0.8
-0
SECOND-ORDER
FIRST ORDER
200 W/TL
-0
-0
-o
-o
U
REFERENCE POINT
0.6
Ix
-_0
E06(-3 TL)
0.4
E05 (-2 TL)
E03 (all)
0.2
0.4
CORRECTED
0.6
0.8
q*/w. - hq
FIG. 7.18 STABILITY MAP IN THE ENTHALPY - SINGLE-PHASE-LENGTH PLANE-SERIES E EXPERIMENTS
1.0
FIG. 8.1 OPEN LOOP BLOCK DIAGRAM OF THE BOILING CHANNEL
6Ap
FIG. 8.2 CLOSED LOOP BLOCK DIAGRAM OF THE BOILING CHANNEL
=
0
1.4
1.2
A
A
1.0 |
-A
-
A
.
D
Wc3
x
ci
0.8
20W/T7
pA
A200 W/TL
0.6
UN4IFORM HEAT FLUX DISTRIBUTION
03 300
W/TL
0.4
SUBCOOLING, ATsubb
At ex= At + At
;
At1 = z /2V
; At2 _5
All zero order threshold and transit ion pts.
[*F]
h dz
, calculated by stepwise
V (z) integration; values of
zobb
V9 from PRESDR results
FIG. 8.3 CORRELATION OF THE PERIOD OF THE OSCILLATION
'3o S
8s
W
w0
Curve
A
B
C
D
1
bb
[lbm/hr] [s]
Point
012-01 Stable
012-92 Threshold
012-03 Unstable
D12-06 Unstable
Subcooling, ATsubb
ZOb/V
T
=
547
410
310
2'6
8.5 to 5.7
o
[s]
2.28
2.44
2.43
0.960+ Plotted values
multiplied by 2
0.805
0.710
0.426
F ; Power, q = 300 W/TL, unif.
The curves are graduated in period [s].
FIG. 8.4 OPEN LOOP TRANSFER FUNCTIONS AT LOW SUBCOOLING
Nib
I
I'Re
iL; 2*6
1~.
A
B
35
Io 5
8
'.5
.7.
1.0,
45
5.5
wW0
Curve
A
Point
D16-12 Stable
zt~
Zb/Voo
t
[lbm/hr]
[s]
[s]
445
-
4.80
B 016-93 Threshold 225
7.75 5.07
C D16-14 Unstable 197
7.75 5.18
Subcooling, ATsubb = 51.5 *F ; Power, qw = 300 W/TL, unif.
The curves are graduated in periods [s].
FIG. 8.5 OPEN LOOP TRANSFER FUNCTIONS AT HIGH SUBCOOLING